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SUMMARY TECHNICAL REPORT 

OF THE 

NATIONAL DEFENSE RESEARCH COMMITTEE 


This document contains information affecting the national defense 
of the United States within the meaning of the Espionage Act, 50 
U. S. C., 31 and 32, as amended. Its transmission or the revelation of 
its contents in any manner to an unauthorized person is prohibited 
by law. 

This volume is classified CONFIDENTIAL in accordance with 
security regulations of the War and Navy Departments because 
certain chapters contain material which was CONFIDENTIAL at 
the date of printing. Other chapters may have had a lower classi- 
fication or none. The reader is advised to consult the War and Navy 
agencies listed on the reverse of this page for the current classifica- 
tion of any material. 


PIQUADED UNCLASSIFIED 


ORDER S£C Lm BY TAG 0 5 2 lj3 


Manuscript and illustrations for this volume were prepared for 
publication by the Summary Reports Group of the Columbia 
University Division of War Research under contract OEMsr-1131 
with the Office of Scientific Research and Development. This vol- 
ume was printed and bound by the Columbia University Press. 

Distribution of the Summary Technical Report of NDRC has 
been made by the War and Navy Departments. Inquiries concern- 
ing the availability and distribution of the Summary Technical 
Report volumes and microfilmed and other reference material 
should be addressed to the War Department Library, Room 
lA-522, The Pentagon, Washington 25, D. C., or to the Office of 
Naval Research, Navy Department, Attention: Reports and 
Documents Section, Washington 25, D. C. 

Copy No. 



This volume, like the seventy others of the Summary Technical 
Report of NDRC, has been written, edited, and printed under 
great pressure. Inevitably there are errors which have slipped past 
Division readers and proofreaders. There may be errors of fact not 
known at time of printing. The author has not been able to follow 
through his writing to the final page proof. 

Please report errors to: 

JOINT RESEARCH AND DEVELOPMENT BOARD 
PROGRAMS DIVISION (STR ERRATA) 

WASHINGTON 25, D. C. 

A master errata sheet will be compiled from these reports and sent 
to recipients of the volume. Your help will make this book more 
useful to other readers and will be of great value in preparing any 


revisions. 


SUMMARY TECHNICAL REPORT OF DIVISION 6, NDRC 


VOLUME 21 


TORPEDO STUDIES 


OFFICE OF SCIENTIFIC RESEARCH AND DEVOLOPMENT 

VANNEVAR BUSH, DIRECTOR 

NATIONAL DEFENSE RESEARCH COMMITTEE 
JAMES B. CONANT, CHAIRMAN 

DIVISION 6 
JOHN T. TATE, CHIEF 



WASHINGTON, D. C., 1946 


NATIONAL DEFENSE RESEARCH COMMITTEE 



James B. Conant, Chairman 
Richard C. Tolman, Vice Chairman 


Roger Adams Army Representative^ 

Frank B. Jewett Navy Representative' 

Karl T. Compton Commissioner of Patents' 

Ir vin Stewart, Executive Secretary 


^Army representatives in order of service: 

Maj. Gen. G. V. Strong Col. L. A. Denson 

Maj. Gen. R. C. Moore Col. P. R. Faymonville 

Maj. Gen. C. C. Williams Brig. Gen. E. A. Regnier 

Brig. Gen. W. A. Wood, Jr. Col. M. M. Irvine 

Col. E. A. Routheau 


~Navy representatives in order of service: 

Rear Adm. H. G. Bowen Rear Adm. J. A. Finer 
Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren 
Commodore H. A. Schade 
^Commissioners of Patents in order of service: 

Conway P. Coe Casper W. Corns 


NOTES ON THE ORGANIZATION OF NDRC 


The duties of the National Defense Research Com- 
mittee were (1) to recommend to the Director of 
OSRD suitable projects and research programs on the 
instrumentalities of warfare, together with contract 
facilities for carrying out these projects and pro- 
grams, and (2) to administer the technical and scienti- 
fic work of the contracts. More specifically, NDRC 
functioned by initiating research projects on recjuests 
from the Army or the Navy, or on recpiests from an 
allied government transmitted through the Liaison 
Office of OSRD, or on its own considered initiative 
as a result of the experience of its members. Pro- 
posals prepared by the Division, Panel, or Committee 
for research contracts for performance of the work 
involved in such projects were first reviewed by 
NDRC, and if approved, recommended to the Direc- 
tor of OSRD. Upon approval of a proposal by the 
Director, a contract permitting maximum flexibility 
of scientific effort was arranged. The business aspects 
of the contract, including such matters as materials, 
clearances, vouchers, patents, priorities, legal mat- 
ters, and administration of patent matters were 
handled by the Executive Secretary of OSRD. 

Originally NDRC administered its work through 
five divisions, each headed by one of the NDRC 
members. These were; 

Division A— Armor and Ordnance 
Division B— Bombs, Fuels, Cases & Chemical Prob- 
lems 

Division C— Communication and Transportation 
Division D— Detection, Controls, and Instruments 
Division E— Patents and Inventions 


In a reorganization in the fall of 1942, twenty-three 
administrative divisions, panels, or committees were 
created, each with a chief selected on the basis of his 
outstanding work in the particular field. The NDRC 
members then became a reviewing and advisory 
group to the Director of OSRD. The final organiza- 
tion was as follows: 


Division 1— Ballistic Research 

Division 2— Effects of Impact and Explosion 

Division 3— Rocket Ordnance 

Division 4— Ordnance Accessories 

Division 5— New Missiles 

Division 6— Sub-Surface Warfare 

Division 7— Fire Control 

Division 8— Explosives 

Division 9— Chemistry 

Division 10— Absorbents and Aerosols 

Division 1 1— Chemical Engineering 

Division 12— Transportation 

Division 13— Electrical Communication 

Division 14— Radar 

Division 15— Radio Coordination 

Division 16— Optics and Camouflage 

Division 17— Physics 

Division 18— War Metallurgy 

Division 19— Miscellaneous 

Applied Mathematics Panel 

Applied Psychology Panel 

Committee on Propagation 

Tropical Deterioration Administrative Committee 


iv 


l.ibrary of Congress 

201 5 460883 



NDRC FOREWORD 


AS EVENTS of the j^ai's preceding 1940 revealed 
more and more clearly the seriousness of the 
world situation, many scientists in this country came 
to realize the need of organizing scientific research 
for service in a national emergency. Recommenda- 
tions which the}" made to the White House were given 
careful and S3"mpathetic attention, and as a result 
the Xational Defense Research Committee [NDRC] 
was formed by Executive Order of the President in 
the summer of 1940. The members of NDRC, ap- 
pointed by the President, were instructed to supple- 
ment the work of the Army and the Navy in the 
development of the instrumentalities of war. A year 
later, upon the establishment of the Office of Scien- 
tific Research and Development [OSRD], NDRC 
became one of its units. 

The Summary Technical Report of NDRC is a 
conscientious effort on the part of NDRC to sum- 
marize and evaluate its work and to present it in a 
useful and permanent form. It comprises some 
seventy volumes broken into groups corresponding 
to the NDRC Divisions, Panels, and Committees. 

The Summary Technical Report of each Division, 
Panel, or Committee is an integral survey of the work 
of that group. The first volume of each group’s re- 
port contains a summary of the report, stating the 
problems presented and the philosophy of attacking 
them and summarizing the results of the research, 
development, and training activities undertaken. 
Some volumes may be “state of the art” treatises 
covering subjects to which various research groups 
have contributed information. Others may contain 
descriptions of devices developed in the laboratories. 
A master index of all these divisional, panel, and 
committee reports which together constitute the Sum- 
mary Technical Report of NDRC is contained in a 
separate volume, which also includes the index of a 
microfilm record of pertinent technical laboratory 
reports and reference material. 

Some of the NDRC-sponsored researches which 
had been declassified by the end of 1945 were of suffi- 
cient popular interest that it was found desirable to 
report them in the form of monographs, such as the 
series on radar by Division 14 and the monograph on 
sampling inspection by the Applied Mathematics 
Panel. Since the material treated in them is not dupli- 


cated in the Summary Technical Report of NDRC, 
the monographs are an important part of the story 
of these aspects of NDRC research. 

In contrast to the information on radar, which is of 
widespread interest and much of which is released to 
the public, the research on subsurface warfare is 
largely classified and is of general interest to a more 
restricted group. As a consequence, the report of 
Division 6 is found almost entirely in its Summary 
Technical Report, which runs to over twenty vol- 
umes. The extent of the work of a Division cannot 
therefore be judged solely by the number of volumes 
devoted to it in the Summary Technical Report of 
NDRC: account must be taken of the monographs 
and available reports published elsewhere. 

Any great cooperative endeavor must stand or fall 
with the will and integrity of the men engaged in it. 
This fact held true for NDRC from its inception, 
and for Division 6 under the leadership of Dr. John 
T. Tate. To Dr. Tate and the men who worked with 
him— some as members of Division 6, some as rep- 
resentatives of the Division’s contractors — belongs 
the sincere gratitude of the Nation for a difficult and 
often dangerous job well done. Their efforts contrib- 
uted significantly to the outcome of our naval 
operations during the war and richly deserved the 
warm response they received from the Navy. In 
addition, their contributions to the knowledge of the 
ocean and to the art of oceanographic research will 
assuredly speed peacetime investigations in this field 
and bring rich benefits to all mankind. 

The Summary Technical Report of Division 6, 
prepared under the direction of the Division Chief 
and authorized by him for publication, not only pre- 
sents the methods and results. of widely varied re- 
search and development programs but is essentially 
a record of the unstinted loyal cooperation of able 
men linked in a common effort to contribute to the 
defense of their Nation. To them all we extend our 
deep appreciation. 

Vannevar Bush, Director 
Office of Scientific Research and Development 

J. B. Con ANT, Chairman 
National Defense Research Committee 



V 



FOREWORD 


One of the most serious obstacles to the use of tor- 
pedoes from aircraft is the likelihood of damage and 
consequent failure of the torpedo if the water-entry 
speed is high. Unless the speed is high, however, the 
attacking plane is a “sitting duck.” This was brought 
home with tragic force in the Battle of ^Midway. 

In July 1943, the Xavy requested NDRC to under- 
take the design of an improved torpedo capable of 
withstanding the shock of water entry when launched 
at aircraft speeds as high as 400 knots. The project 
was assigned to Division 6 and thus initiated the stu- 
dies reported in this volume. 

It was soon discovered that the addition of a sim- 
ple ring to the tail fins of the standard Mark 13 tor- 
pedo brought its performance up to the limiting speed 
of available torpedo planes. With this discovery, the 
pressure for immediate improvement with respect to 
water entry was further increased, and at the request 
of the Navy the project was broadened to include a 
study of every aspect of torpedo design. As a result 
of this comprehensive program, not only was a tor- 
pedo developed which fully met all requirements, but, 
of longer-term interest, an analysis was furnished of 
the basic physical factors affecting the overall design 
of any future torpedo. 

The success of this project was made possible only 
by the cordial and effective cooperation of many 
individuals and agencies. 

The technical program of the Division was carried 
out under contracts with Columbia University, Cali- 
fornia Institute of Technology, Massachusetts Insti- 
tute of Technology, and the American Can Company. 


The N’avy, through the Bureau of Ordnance and 
the Naval Torpedo Station at Newport, not only 
provided test facilities, but made freely available the 
knowledge and skills gained through their long ex- 
perience in design, production, and use of torpedoes. 
And at the suggestion of the Commanding Officer a 
project engineer was permanently stationed at New- 
port to assure complete and continuous exchange of 
information. 

The Army, because of its interest in a related pro- 
gram, maintained close liaison throughout the dura- 
tion of the project. 

Division 3 of NDRC made possible crucial full- 
scale tests of high-speed water entry by providing 
through their contract with California Institute of 
Technology both personnel and facilities. 

Division 7 of NDRC contributed consulting ser- 
vices on the control problem. 

The General Electric Company gave to the groups 
working on power plant design the full advantage of 
their knowledge and experience in the design and 
construction of high-temperature gas turbines. 

To all of these individuals and agencies, and in par- 
ticular to Dr. W. V. Houston, Director of the Colum- 
bia University Special Studies Group who served as 
director of the entire program, the Division makes 
grateful acknowledgment. 

John T. Tate 
Chief, Division 6 



• • 

Vll 



i 


PREFACE 


This report was prepared by the Columbia Univer- 
sity Special Studies Group as part of the studies made 
in connection with Project NO-176. It covers essen- 
tially the theoretical studies of torpedo performance 
and the corresponding indications as to proper tor- 
pedo design. 

Although some of the theoretical work presented 
was done by the Special Studies Group, much of this 
report is a compilation of experimental results and 
theoretical developments carried on by various 
groups at a number of places. When possible, an 
attempt has been made to give credit to these groups 
for the work that they have done. 

The work on the theory of torpedo control as dis- 
cussed in this report is largely due to Dr. L. I. Schiff, 
while the studies associated with air flight and water 
entry have been compiled and developed by Marvin 
Gimprich. 


W. V. Houston 



I 


CONTENTS 


PART I 

GENERAL DISCUSSION AND SUMMARY 


CHAPTER PAGE 

1 General Requirements for Torpedoes 3 

2 Elements of a Torpedo 6 

3 Outline and Summary 10 


PART II 

HYDRODYNAMICS AND AERODYNAMICS 


4 

5 

6 
7 


Hydrodynamic and Aerodynamic Forces and Moments . 

Air Flight of a Torpedo 

Water Entry 

Underwater Run 


19 

21 

50 

114 


PART III 

CONTROL SYSTEMS 


8 General Discussion of Controls 123 

9 Proportional Control 129 

10 Two-Position Control 135 

11 Steering Control 138 

12 Depth Control 143 

13 Power Plant 149 

Bibliography 151 

Contract Numbers 155 

Service Project Numbers 156 

Index 157 







XI 




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PART I 


GENERAL DISCUSSION AND SUMMARY 




Chapter 1 

GENERAL REQUIREMENTS FOR TORPEDOES 


T he term ‘‘torpedo’^ has been used at various 
periods in the history of modern warfare to mean 
a variety of weapons, but nearly always these have 
contained a charge of high explosive. In the naval 
warfare of the past 50 years the term has come to be 
applied almost exclusively to the automobile torpedo. 
This is a self-propelled and self-steered underwater 
vehicle that carries a charge of explosive to a point at 
which it will explode against the underwater part of 
an enemy ship. The principal problems of torpedo de- 
sign can then be classified as those associated with its 
launching, its propulsion, its steering, and the means 
for exploding the charge. 

The philosophy of torpedoes has always involved 
the idea that the explosive charge carried should be 
such as to cause decisive damage to the enemy. The 
rest of the torpedo, the power plant and the control 
mechanism, has been so elaborate and so expensive 
that economy of effort could be obtained only if one 
torpedo hit was sufficient to sink a ship. In fact, the 
fraction of the weight of a torpedo made up of the 
explosive charge is usually only somewhere between 
0.2 and 0.3 so that the equipment necessary to con- 
vey the charge to its objective is much more elaborate 
than the charge and its firing mechanism. 

In recent years, however, defenses against torpe- 
does have been built into all capital ships and many 
smaller ones so that the probability of sinking such a 
ship with a single torpedo is not very great. Further- 
more, even cargo ships are so divided into compart- 
ments that usually a single torpedo will not be suffi- 
cient to sink one of them. It appears that a very large 
increase in explosive charge may be necessary ma- 
terially to increase the probability of sinking from a 
single hit. For this reason, it may be more economical 
of effort to count on using two or more automobile 
torpedoes to sink even a merchant ship, but it is im- 
probable that the complication of torpedo mechan- 
ism should ever be used to transport anything less 
than a very seriously damaging charge of explosive. 

Another characteristic of torpedoes is that the ex- 
plosion occurs against the part of the ship that is 
underwater. This is clearly more damaging than a hit 
on the superstructure, not only because a hole be- 


neath the water line will cause the ship to lose 
buoyancy, but also because the presence of the water 
increases the effectiveness of the explosion. Modern 
battleships are so protected by armor plate in the 
neighborhood of the water line and for some distance 
below it that hits by shells in this region are con- 
siderably reduced in effectiveness. A hit by a torpedo 
on this armor belt will normally cause very little 
damage. Nevertheless, this kind of armor cannot be 
used to cover the entire hull for the buoyancy of the 
ship would be seriously reduced by it. A properly 
adjusted torpedo can hit the skin of the ship below 
this armor belt and do serious damage. For this 
reason a torpedo is a formidable weapon against any 
ship. 

The automobile torpedo is normally fired at a con- 
siderable distance from its target. It normally travels 
in the water at a speed greater than that of the 
fastest ships. Nevertheless, even this speed is suffi- 
ciently low to make it necessary to predict the posi- 
tion of the target some time in advance. A torpedo 
traveling at 45 knots will require some 4 minutes to 
travel 6,000 yd. During this time, a ship making 30 
knots will travel 4,000 yd. In case the firing of the 
torpedo could be detected, it would be quite possible 
for the target ship to avoid being hit by turning 
sharply. However, torpedoes can be launched in 
many cases without this launching being detected, 
and more often a number are launched almost simul- 
taneously so that it is difficult for the target to avoid 
them all. 

In the case of torpedoes used in naval engagements, 
long ranges are frequently used, and some torpedoes 
may need a range as great as 20,000 yd. It is clear, 
however, that no single aimed torpedo can be effec- 
tive at such a range unless the target continues on a 
steady course throughout the full torpedo run. For 
such reasons, it is important to reduce the running 
time of a torpedo as much as possible. Apparently the 
most effective way to do this would be to increase the 
speed of the torpedo, but, since this is very difficult 
to accomplish, consideration must also be given to 
the possibility of reducing the length of the under- 
water run. 




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3 




4. 


GENERAL REQUIREMENTS FOR TORPEDOES 


1.1 SPEED AND RANGE OF SUBMARINE 

AND DESTROYER LAUNCHED 

TORPEDOES 

The torpedo is the ideal weapon for a submarine 
vessel. It can be fired in many cases before the enemy 
is aware of the presence of the submarine. If an 
electric torpedo, or some other torpedo without a 
visible wake, is used, the first warning of an attack 
may come from the explosion. Under these conditions 
a long range may be useful, and certainly a high tor- 
pedo speed is useful in increasing the aiming accur- 
acy. Other things being equal, it is probable that 
more hits can be made with a high-speed than with a 
low-speed torpedo. On the other hand, speed can be 
obtained only by a sacrifice of other qualities and 
principally by a sacrifice of range or by a considerable 
increase in size and weight. Hence it is important to 
evaluate carefully the minimum useful range from 
which a torpedo can be fired and to emphasize its 
use. This would improve the accuracy of aiming, not 
only because of the shorter range and the reduced 
effect of aiming angle errors, but also because the 
maximum speed could then be obtained from the fuel 
carried. 

1.2 SPEED AND RANGE OF AIRCRAFT 

TORPEDOES 

In the case of aircraft torpedoes, the situation is 
somewhat different. The launching of the torpedo 
can usually be observed by the target, and the proper 
evasive action can be taken. It appears impractical 
to try to produce sufficient underwater speed in a tor- 
pedo to make this evasive action ineffective if much 
underwater travel is necessary. Since, however, the 
torpedo is dropped from an airplane, advantage can 
be taken of the speed of the plane and the altitude 
from which the torpedo is dropped to provide a long 
air travel. The speed of this travel in air is so much 
greater than any feasible underwater speed that most 
of the distance between the point of release and the 
target should be covered in the air travel. The prin- 
cipal requirement appears to be the development of a 
torpedo that can be launched from very high-speed 
planes and from high altitudes, with a sufficient 
power plant to travel a short distance under water at 
a moderate speed. Along with such a torpedo ade- 
quate torpedo directors must be provided. 

If a torpedo can be released from 6,400 ft at 350 
knots so as to strike the water 500 yd from its target. 


it will spend only about 20 sec in the air, traveling 
almost 4,000 yd. If its underwater speed is 40 knots, 
the 500 yd will require about 23 sec so that a total 
distance of nearly 4,500 yd can be covered in about 
43 sec. This is not time enough for much manuevering 
on the part of the target. 

13 REQUIREMENTS FOR STEERING 
AND DEPTH CONTROL 

The specifications laid down for the steering and 
depth control of a torpedo will depend on the way in 
which it is to be used. It is necessary that the depth 
control be adequate to get the explosive to the desired 
depth on the target. When contact exploders are to 
be used and the target is a battleship, the depth 
mechanism must insure that the torpedo hits below 
the armor belt, and it must not fail to keep the tor- 
pedo from running completely under the ship. When 
influence exploders are used, the necessary accuracy 
of depth-keeping is not so great since the exploder 
will operate at any depth down to some distance be- 
low the keel. 

The accuracy of steering required depends upon 
the length of the underwater run. The deviation from 
the set course must not be so great as to dominate 
other errors, but there is also no need to require of 
the steering equipment that it maintain its direction 
for a time longer than the torpedo will run or with an 
accuracy far greater than the accuracy of aiming that 
is possible. In general, it may be expected that a tor- 
pedo with a long underwater run will require greater 
accuracy than one with a short underwater run. 
Thus a submarine-launched torpedo with a run of 
some 5,000 yd would be expected to require roughly 
five times the steering accuracy of an aircraft torpedo 
with an underwater run of only about 1,000 yd. 
Nevertheless, the possible evasive action that may be 
taken by a target attacked by a long range torpedo 
and the use of salvos of long range torpedoes may 
again reduce the value of high accuracy steering in 
even these cases. Since the requirements of steering 
accuracy present some of the most troublesome prob- 
lems in torpedo manufacture and maintenance, it is 
important to limit such requirements to those which 
really contribute to the effectiveness of the torpedoes. 

In the case of an aircraft torpedo launched from a 
high altitude for a short underwater run it is neces- 
sary that: (I) the torpedo does not dive so deep as to 
strike the bottom and stick or be damaged; (2) the 
torpedo recover from its set depth and arm before 


REQUIREMENTS FOR STEERING AND DEPTH CONTROL 


5 


reaching the target. These conditions, which are de- 
pendent only in part on the operation of the depth- 
control mechanisms, make necessary a complete un- 
derstanding of the water-entry phenomena in design- 
ing such a torpedo. To a considerable extent, the 
water-entry behavior is determined by the external 
shape and the distribution of mass, as well as the 


torpedo orientation at entry. 

The studies described in this report include many 
things common to all torpedoes, but particular em- 
phasis has been placed on these problems of air travel, 
water entry, and control that are important in the 
design of aircraft torpedoes and, in particular, in the 
design of the Mark 25 aircraft torpedo. 


Chapter 2 

THE ELEMENTS OF A TORPEDO 


A TORPEDO may be considered as being composed of 
. several more or less independent elements prop- 
erly balanced with one another to make a well- 
integrated weapon. For convenience of discussion, 
the general principles and basic underlying features 
of each of the elements can be developed separately. 

2.1 EXTERNAL SHAPE 

Torpedo bodies are usually cylindrical in shape and 
are provided with a rounded nose that may be either 
hemispherical or pointed, a tail which supports fins 
for stability, and rudders and elevators for steering 
and depth control. 

This general shape has been found most suitable 
to furnish the volume required to contain the ex- 
plosive charge, the power plant and control mecha- 
nism, and to provide for the dynamic stability and 
minimum drag necessary for proper behavior in 
water. These latter requirements are discussed in 
detail in Chapters 4 and 7. 

These shape requirements hold for submarine- 
launched torpedoes and, in most cases, for torpedoes 
launched from surface vessels. For an aircraft- 
launched torpedo, however, the shape must be also 
suitable for proper travel through air and for proper en- 
try into the water. In general, the shapes required for 
the three different phases of travel, air travel, water 
entry, and controlled underwater travel, are differ- 
ent. Stability in air travel is provided by equipping 
the torpedo with light wood appendages, which 
break off with entry of the torpedo into the water. 
These air stabilizers are of various types. Some are 
mounted on the tail, while others, such as the drag 
ring or “pickle barrel,” are on the nose. 

In most cases it has been possible to select a shape 
that is both adequate for water entry and satisfac- 
tory for the underwater run. However, various ap- 
pendages have been tried, such as spoiler rings and 
Townend extensions to the nose, that are designed 
to stay on during water entry and to detach only 
when the steady run begins. It is probable that opti- 
mum control of the water-entry phase would recpiire 
appendages on the tail as well as those tliat have been 
tried on the nose. Nevertheless, it has been found 
possible to get adecpiate performance, under certain 


conditions, by means of a shape that is also suitable 
for steady running. 

2.2 the explosive charge and 

THE EXPLODER 

The size and nature of the explosive charge for a 
torpedo depend to a considerable extent on the kind 
of targets for which it is designed. Most torpedoes 
carry charges in the neighborhood of 600 to 800 lb 
of Torpex. This charge appears to be adequate to 
damage a battleship and to cause very serious dam- 
age to lighter vessels. It appears that an increase of a 
few hundred pounds would make relatively little 
difference, and, unless it were decided to multiply the 
charge by a factor of at least 3 to 5, it would be hardly 
worth increasing at all. 

An essential part of the torpedo warhead is the 
exploder mechanism. Exploder mechanisms are now 
available that will detonate the charge in the vicinity 
of the target rather than depending on an impact. 
This makes possible the use of greater depths and so 
minimizes the danger of hitting the armored belt of a 
capital ship. 

The torpedo studies included in this report do not 
include studies of the explosive and the exploder 
mechanism. 

2.3 PROPULSION MECHANISM 

The propulsion mechanism occupies the major part 
of the torpedo both in volume and in weight. For this 
reason, the overall size is determined very largely by 
the requirements placed on speed and range. In order 
to get the maximum speed and range with a given 
weight, it is necessary that the power plant operate as 
efficiently as possible. The power necessary to drive 
the torpedo is approximately proportional to the 
third power of the speed at which it runs, provided 
the propulsive efficiency is independent of the speed. 
For the conventional propeller-driven type of torpedo, 
this is approximately true, but for some of the pro- 
posed types of drive, the efficiency increases rapidly 
with the speed so that the power required does not 
increase as rapidly as the third power of the speed. 

The total energy recpiired for a torpedo is propor- 


6 


CONTROL MECHANISMS 


7 


tional to the second power of the speed and to the 
range. Consequently, the amount of fuel necessary 
is similarly proportional to these quantities. A tor- 
pedo differs from a surface ship in that it must carry 
not only its fuel in the ordinary sense but also the 
oxidant. In most torpedoes, the oxidant has been in 
the form of compressed air. In fact, the original tor- 
pedoes were driyen by compressed air alone. Later, a 
little fuel was added to warm up the air so as to make 
it more effectiye, and finally the process became one 
of complete combustion. For the purpose of combus- 
tion it is clear that compressed oxygen would be 
better than compressed air because the air is only 
about 30 per cent oxygen. Xeyertheless, the hazards 
associated with' the use of compressed oxygen haye 
preyented its extensiye adoption. 

With either compressed air or compi-essed oxygen 
a heayy tank is necessary to contain the gas. In fact, 
approximately 4 lb of steel is necessary to contain 
1 lb of air. For this reason, obyiously it would be 
better if the oxidant could be carried in the form of a 
liquid since only about 1 lb of container is necessary 
for 1 lb of liquid. This 60 per cent reduction in the 
weight of the fuel and its container is a yery impor- 
tant improyement in torpedo power plants, and such 
systems are now being deyeloped. Probably the most 
adyanced is that in which liquid hydrogen peroxide 
is used as the oxidant. On its way to the combustion 
chamber this is decomposed into oxygen and water. 
The oxygen combines with the fuel, and the water 
cools the flame to a point at which it can be used in 
the turbine. 

Any such improyement in the weight of the fuel 
and its containers can be translated directly into tor- 
pedo range. If the use of hydrogen peroxide reduces 
the fuel and container weight by 60 per cent, it pro- 
yides a means of increasing the range of a torpedo by 
a factor of 2.5 without increasing its weight. Whether 
the saying is put into increased range or into in- 
creased explosiye depends on which of the two is re- 
garded as the more important. 

Propulsion mechanisms will not be discussed ex- 
tensively in this report. An extensiye study of them 
has been made by the X"ayy Department to be used 
as a basis for future torpedo deyelopment. 

2.4 CONTROL MECHANISMS 

The conyentional automobile torpedo runs on a 
preyiously set course at a predetermined depth. The 


depth-control mechanism and the steering mecha- 
nism are to a large extent independent. 

The operation of these mechanisms must be closely 
associated with the hydrodynamics of the torpedo 
body. This is particularly true of the pendulum, in 
which care must be taken that the natural period of 
the pendulum is not too close to the period at which 
the torpedo tends to oscillate in depth. Furthermore, 
the stability of the control depends not only on the 
control mechanism itself but also on the hydrodyna- 
mic constants of the body. 

The deyelopment of torpedo-control mechanisms 
can then be diyided into two parts. In the first part, 
it is important to measure the hydrodynamic con- 
stants of the torpedo and then to determine the prop- 
erties of the control necessary to operate in the de- 
sired manner. In particular, it is important to deter- 
mine the maximum time lag that can be permitted 
between the actuation of the mechanism and the final 
operation of the rudder and eleyator. This is inyersely 



Figure 1. Estimated curves showing the speed and 
range attainable for various total weights and an ex- 
plosive charge of 300 lb. 


8 


THE ELEMENTS OF A TORPEDO 


proportional to the velocity, and its value for any 
given torpedo depends on the hydrodynamic con- 
stants. 

The second part concerns the design of mechanisms 
having suitable properties. The present report is con- 
cerned only with the theoretical analysis. The 
progress that has been made under Project NO- 176 


between the desire to have maximum weight of ex- 
plosive, speed, and range and the desire to have a 
minimum overall weight. 

In a very rough way the weight of a torpedo may 
be regarded as made up of four parts. These are E, 
the weight of the explosive; S, the weight of the 
external shell; F, the weight of the fuel and its con- 



Figure 2. Estimated curves showing the speed and 
range attainable for various total weights and an ex- 
plosive charge of 700 lb. 



Figure 3. Estimated curves showing the speed and 
range attainable for various total weights and an ex- 
plosive charge of 1,500 lb. 


in the design of depth and steering mechanisms is 
reported in connection with the design of the Mark 
25 torpedo. 

2.6 WEIGHT AND SIZE OF THE 

TORPEDO 

The outstanding properties of a torpedo that must 
be correlated with each other are (1) range, (2) 
speed, (3) weight of explosive, and (4) overall weight. 
These four quantities are all related, and in the de- 
sign of a torpedo suitable compromises must be made 


tainers ; and P, the weight of the turbine, gear train, 
and propeller system. The weight of the shell is 
roughly proportional to the external surface of the 
torpedo and hence may be regarded as proportional 
to the two-thirds power of the total weight. Hence we 
may set S = ylF^l 7 is a constant that depends upon 
the material of which the shell is made, upon its 
thickness, and upon the shape of the torpedo. 

The amount of fuel necessary is proportional to 
(1) the range of the torpedo, (2) the square of the 
velocity, and (3) the external surface. The amount of 
fuel necessary is proportional to the external surface 




WEIGHT AND SIZE OF THE TORPEDO 


9 


because the drag is proportional to the external sur- 
face and hence to the two-thirds power of the weight. 
Hence we ma}" accept F = The constant a 

depends upon the fuel system used and will be very 



Figure 4. Curves showingt he estimated speed and 
range attainable for various total weights with the use of 
a liquid oxidant and an explosive charge of 700 lb. 


much smaller in case hydrogen peroxide is used than 
when compressed air is used. 

In a very rough way, the weight of the power plant 
may be set proportional to the cube of the speed and 
to the external torpedo surface. Hence we may regard 
P = 

Since the total weight of the torpedo is the sum of 
these four parts, it is possible to write an equation 
connecting the total weight of the explosive, the 
speed, and the range as follows : 


ir = E {y av’^R) . 

The constants, a, (3, and y must be determined from 
some known torpedo. If they are evaluated for the 
Mark 25 torpedo, the values are approximately 

a = 7.00 X 10-7 , 
d = 2.45 X 10-^ , 
y = 4.00 . 



Figure 5. Total w'eight as a function of the weight of 
explosive for three cases. The power plant is assumed to 
be similar to that of the Mark 25. 


By using these constants, it is possible to plot rela- 
tionships between the various quantities in the equa- 
tion. In particular. Figures 1, 2, and 3 show the speeds 
and ranges that can possibly be attained with given 
overall weight. Other curves present these data in 
slightly different ways. 

These curves point out clearly the cost at which 
high speed and long range are attained. In most tor- 
pedoes, the explosive is a relatively small part of the 
total weight, while the fuel and power plant make up 
most of it. If the diameter is increased it is possible 
to increase the explosive charge with only a small 
increase in overall weight, but doubling the range or 
the speed makes a very serious change in the torpedo. 


Chapter 3 

OUTLINE AND SUMMARY 


3.1 AIR TRAVEL 

AX AIRCRAFT TORPEDO is launched by dropping it 
from an airplane that is traveling at a high rate 
of speed, and the most important line of development 
of this weapon is concerned with increasing the speed 
and the altitude from which it can be launched. The 
initial travel of the torpedo through the air is equally 
important with its underwater travel, not only be- 
cause the major part of the distance may be traversed 
in the air, but also because the torpedo must end its 
air travel in such a way as to make a proper entry 
into the water. The importance of this proper entry 
into the water is easily understood from the analogy 
between the torpedo and a diver. As is well known, a 
diver can safely dive from an elevation of 100 ft if he 
enters the water properly, while he can be seriously 
injured if he enters the water flat from a veiy much 
lower altitude. 

The early experiments in launching torpedoes from 
aircraft Avere confined to dropping them from only 
a short distance above the water, such as 30 to 50 ft, 
and at speeds much less than 100 knots. These ex- 
periments emphasized the necessity for very rugged 
construction in order to withstand the shock of water 
entry. It appears, however, that in addition serious 
damage to the torpedo can only be minimized by 
providing for a clean entry. In fact, some evidence 
points to the fact that a 15-ft horizontal drop of a 
torpedo, such as has been frequently used for proof 
launchings, may be more damaging than a clean 
entry at speeds of over 200 knots. 

For a clean entry it is necessary that the torpedo 
be traveling parallel to its axis at the time of impact 
with the water, and this requires proper stabilization 
of the torpedo in the air. Due to the action of the air 
forces, a simple cylindrical body tends to set itself 
perpendicular to its direction of travel, and a torpedo 
has somewhat the same tendency. If a bare torpedo 
is launched from a considerable elevation, it will 
enter the water flat and will suffer severe damage. To 
overcome this tendency, a large tail is required. Since 
this kind of tail is undesirable for the underwater 
run, it is constructed of light wood so as to break off 
on water entry. 

A wide variety of stabilizers has been tried. Present 


practice in the United States Navy is to use a Mark 
2-1 stabilizer combined Avith a drag ring or “pickle 
barrel” on the nose. The Mark 2-1 stabilizer is a 
simple AAmoden box that is slipped over the torpedo 
fins. The Mark 1 drag ring is a short AAmoden cylinder 
that slips OA^er the nose. The combined effect of these 
tAvo auxiliary devices is to cause the torpedo to be 
stable AATen traveling parallel to its air trajectory. 
The stabilizers also introduce sufficient damping so 
that during its air flight the torpedo aatU oscillate 
around its position of equilibrium AAuth decreasing 
amplitude. If, then, the air flight is long enough, 
initial disturbances that are present due to conditions 
at release may be damped out, and the torpedo can 
enter the AA^ater smoothly and cleanly. This fact sug- 
gests that it is even more desirable to drop a torpedo 
from a high altitude than from a Ioav altitude be- 
cause a longer time is available in Avhich this damping 
action can be effected. 

A more important fact is that the damping effect 
of the stabilizers increases AAdth the speed so that a 
high-speed launching may tend to be cleaner than 
one of loAA^er speed. The damping is proportional 
to the speed, but the time of fall A^aries only as the 
square root of the altitude. Both high speed and 
high altitude therefore, are conducive to a good entry 
in the absence of a Avind. These stabilizers also intro- 
duce a certain amount of drag so that there are some 
altitudes, depending on the release speed, from Avhich 
the torpedo enters the AATiter AAuth a total velocity 
that is e\"en less than the initial release velocity. For 
a loAA^-speed launching this is only at a Ioav altitude, 
but for a high-speed launching this may be true up to 
over 2,000 ft. 

Other stabilizers have been used Avith more elabo- 
rate objectives. The British M.A.T. IV stabilizer is 
intended to cause the torpedo to strike the Avater 
slightly nose-up Avith respect to its trajectory. This 
can be easily accomplished by suitable shaping of the 
tail pieces, but it requires that roll in the air be pre- 
vented. To this end, the British M.A.T. IV stabilizer 
is a very elaborate device, incorporating a gyroscope 
and movable ailerons to correct any tendency to roll. 
Such elaboration is surely necessary if it is desired to 
make the torpedo enter in any other Avay than 
parallel to its trajectory. HoAvever, it Avould seem 


10 


WATER ENTRY 


11 


desirable to construct the torpedo so that entry 
parallel to its trajectory is adequate and so that very 
simple stabilizing devices can be used. 

It is obvious that stabilizing appendages, such as 
have just been mentioned, can be effective only in 
causing the torpedo axis to remain parallel to its 
trajectory with respect to the air. If there is a wind, 
the trajectory with respect to the air will not be 
identical with the trajectory as seen from the ground, 
and it is the trajectory as seen from the ground that 
determines the water-entry conditions. As a conse- 
quence, a torpedo stabilized on its trajectory, and 
traveling with a tail wind, tends to enter the water 
effectively nose-down. A similar torpedo traveling in 
a head wind tends to enter effectively nose-up, and a 
torpedo traveling in a cross wind will enter with a 
certain amount of yaw. These effects of wind cannot 
be overcome b}^ simple air stabilizers, and the}" can- 
not be neglected because they will affect the under- 
water behavior of the torpedo in a significant way. 
However, they can be recognized and understood, 
and proper allowance can be made for them in the 
tactical methods that are used. 

In addition to making the torpedo stable on its 
trajectory, the stabilizers introduce a certain amount 
of drag. The iMark 2-1 stabilizer and the Mark 1 drag 
ring when used on the Mark 13 torpedo introduce 
enough air resistance so that the torpedo may enter 
the water at a speed somewhat less than the speed at 
which it is released. iMost of this drag is due to the 
drag ring itself and not to the tail stabilizer. The 
analysis shows that the resultant speed of the torpedo 
first decreases because of the air resistance and then 
increases because of the acceleration of gravity so 
that it passes through a minimum. This increase does 
not continue indefinitely because the terminal ve- 
locity appears to be between 800 and 900 ft per sec. 
At this speed the air resistance is just equal to the 
weight so that there is no further acceleration. Be- 
cause of this air resistance, the horizontal travel of 
the torpedo may be significantly less than that calcu- 
lated when the air resistance is neglected, and the 
entry angle will also be different. This merely means 
that the more complete calculations must be used for 
predictions of the air flight. 

It is clear that stabilizers could be designed to have 
increased drag and to reduce the velocity at impact 
to a low value. This, however, would result in a short 
horizontal range in air. 

A detailed account of the air trajectory is given in 


Chapter 5, where the necessary properties of the 
stabilizers are analyzed. 

3.2 WATER ENTRY 

The water-entry phase of the aircraft torpedo tra- 
jectory is less subject to theoretical analysis than 
either the air travel or the underwater travel. At the 
beginning of the present study of the problem almost 
no information was available as to the essential 
features of the behavior of the torpedo during this 
stage. The air trajectory is subject to fairly complete 
theoretical analysis, and only certain constants need 
to be evaluated in order to apply the theory to any 
specific case. The same thing is true of the underwater 
run. The water entry, however, presents additional 
complications due to the presence of the water sur- 
face and the vastly different properties of the air and 
the water. For this reason, it has been possible only 
to establish a more or less phenomenological descrip- 
tion of the torpedo behavior and to indicate certain 
convenient terms in which the initial underwater tra- 
jectory can be described. These are satisfactory for 
the water entry of the Mark 13 torpedo and the Mark 
25 torpedo and presumably for other similar shapes. 

When the torpedo nose first strikes the water, the 
torpedo experiences a very high force lasting over a 
very short time. It has not been possible to determine 
with any degree of reliability the exact magnitude or 
duration of this impact force. It is possible, however, 
to give some indication as to the total impulse asso- 
ciated with it. This produces both a sudden change 
of longitudinal velocity and a sudden access to angu- 
lar velocity about a horizontal axis. The sudden 
change in linear velocity is a small fraction of the 
total velocity and seems to play no significant part 
in determining the subsequent torpedo behavior. The 
sudden access to angular velocity, however, is in such 
a direction as to cause the nose to rise and the tail to 
fall and determines whether the subsequent trajec- 
tory turns upward or downward. After the initial 
impact, the torpedo creates a cavity in the water, 
roughly conical in shape, so that only the nose is in 
contact with the water. This state continues until the 
torpedo is several lengths under the surface, and dur- 
ing this time the torpedo is subject to a decelerating 
force that can be described in terms of a drag coeffi- 
cient and is proportional to the square of the velocity. 
The drag coefficient on a hemispherical nose is, at 
this stage, of the order of magnitude of 0.28. 



12 


OUTLINE AND SUMMARY 


The drag force on the torpedo when it is in the 
cavity, with its nose alone in contact with the water, 

K/ 7 7 

is opposite in direction to the torpedo motion but 
does not usually act directly through the center of 
mass. If the torpedo is nose-down to its trajectory, 
this retarding force tends to turn it more nose-down 
and hence tends to overcome the initially produced 
nose-upward angular velocity. If the torpedo is nose- 
up to its trajectory, the retarding force adds to the 
initially produced angular velocity. As a consequence 
of this angular velocity about a horizontal axis, the 
tail of the torpedo will eventually strike either the 
top or bottom of the cavity. If it strikes the top of 
the cavity, the torpedo will travel in a roughly circu- 
lar path concave-downward. If it strikes the bottom 
of the cavity, the path will be concave-upward. Which 
of these two things occurs depends upon the magni- 
tude of the initially acquired angular velocity and 
the later angular acceleration that either adds to or 
subtracts from it. These depend on the entry pitch 
and trajectory angles. There can also occur an inter- 
mediate state in which the torpedo continues to 
travel for some distance without striking either side 
of the cavity. 

If the torpedo enters directly along its trajectory, 
the initially acquired angular velocity will cause the 
nose to rise, and the subsequent retarding force will 
cause it to rise still farther. Under these conditions, 
the tail will strike the bottom of the cavity, and the 
trajectory will curve upward. If the torpedo enters 
slightly nose-down to its trajectory, the suddenly 
acquired angular velocity may be just enough to 
overcome the nose-down angular velocity produced 
by the retarding force. The nose-down pitch angle at 
which this occurs may be designated as the critical 
pitch angle and is found to be something over 2 de- 
grees for the Mark 13 torpedo. If the torpedo enters 
much more nose-down than this critical angle, the 
nose-down turning produced by the retarding force 
will dominate the situation, and the tail of the tor- 
pedo will go to the top of the cavity. This produces 
a down-turning trajectory, and the torpedo will dive 
deep. 

The forces producing the suddenly acquired angu- 
lar velocity and the dependence of these forces on the 
entry conditions depend on the shape of the torpedo 
nose. Indeed, while for most nose shapes the torpedo 
receives an upward impulse, there are noses which 
are sufficiently blunt so that the torpedo nose re- 
ceives a downward impulse at entry and hence the 
suddenly acquired angular velocity will be nose- 


downward in direction. For blunt noses, which are 
rarely used on projectiles entering water designed for 
an underwater run (except for antisubmarine weap- 
ons), the previous discussion must be somewhat 
modified in a manner indicated in Section 6.3. For 
the finer shaped noses which were implicitly assumed 
in the previous discussion, since the angular velocity 
acquired at impact depends on the nose shape, it is 
found that the critical pitch angle will vary with the 
nose shape. 

After a moderately well-defined distance along the 
trajectory, the cavity will close in behind the tor- 
pedo, and the normal hydrodynamic forces will begin 
to act. Except for the continuing retardation, the 
behavior of the torpedo can then be described in the 
terms used for describing its steady underwater run. 

It appears that the major features of the initial 
trajectory are determined before the cavity closes so 
that these major features are not influenced by the 
movement of the torpedo rudders or elevators and, 
consequently, are not dependent on the depth-con- 
trol mechanism or the steering device. These mecha- 
nisms are important, however, in determining the 
trajectory after cavity closure. The hooks to the right 
or left, as well as the maximum depth of dive, are in- 
fluenced by the behavior of the depth mechanism, 
the extent to which the elevators are reduced in 
effectiveness by a shroud ring and the extent to which 
the torpedo heels over. It appears also that a major 
part of the effect of a shroud ring on water-entry 
behavior is due to its shielding of the elevators and 
the corresponding reduction in curvature of the path. 
If the steady-state hydrodynamic constants of the 
torpedo are known, the trajectory after cavity col- 
lapse may be calculated (see Section 6.5). 

On the basis of this kind of a picture of the initial 
underwater trajectory and the approximate determi- 
nation of some of the constants involved, it is possible 
to predict with some degree of certainty the initial 
underwater behavior of the Mark 13 torpedo. Pre- 
sumably the same can be done for the Mark 25, but 
this has not yet been thoroughly tested and verified. 
The situation is described in detail in Chapter 6, 
where equations are given that permit the computa- 
tion of the expected depth of dive for various entry 
conditions. In this work the effect of wind is shown 
to be of great importance. Launching the torpedo in a 
tail wind can easily provide sufficient nose-down 
pitch so that the torpedo dives deep and may well 
strike the bottom in shallow water. Launching in a 
head wind may cause the torpedo to enter with a 




CONTROL OF UNDERWATER RUN 


13 


nose-iip pitch such as to cause excessive broaching 
or to cause excessive torpedo damage on water entry. 

By analogy with the above argument, it can be 
seen that, when the torpedo enters the water with a 
yaw either nose right or left, it will tend to move in 
a more or less circular trajectory to the right or to 
the left. This will introduce the initial hooks that are 
frequently observed. Although there are other more 
important reasons for hooks, it appears probable that 
a cross wind from the left will tend to make a torpedo 
hook to the left and a cross wind from the right will 
tend to make it hook to the right. In other words, 
there is a tendency to hook into the wind. 

It would be highly desirable to find some shape of 
torpedo less sensitive than that of the Mark 13 to 
pitch angle and yaw angle at entry, and it seems quite 
possible that a proper combination of nose shape and 
large tail structure may do this. Nevertheless, there 
is at present no convincing evidence that any shape is 
significantly better than that of the ^lark 13 torpedo 
with the shroud ring or that of the Mark 25 torpedo. 

3.3 CONTROL OF UNDERWATER 

RUN 

An aircraft torpedo, as well as any other kind of 
automobile torpedo, is equipped with steering and 
depth-control mechanism to make it travel on a pre- 
scribed course at a fixed depth. Since many torpedoes 
may travel for a considerable distance through the 
water, the steering performance is of importance in 
determining the probability of hitting the target. The 
depth-keeping performance is even more critical be- 
cause it is desired to set the torpedo to strike the hull 
of a battleship between the lower edge of the armor 
belt and the bottom of the ship. For these reasons, 
the design of a torpedo requires particular attention 
to this steering and depth-control mechanism. 

3 . 3.1 Hydrodynamic Stability 

The studies that have been made in this connection 
have led to a fairly satisfactory theory of torpedo 
control. It has been shown that the behavior of the 
torpedo is the result not only of the control mecha- 
nism itself but also of the hydrodynamic character- 
istics of the torpedo body. These two things can be 
treated more or less separately, and the theory is now 
such that the necessary properties of the control can 
be fairly well specified when the hydrodynamic be- 
havior of the body is known. 


Some of the properties of a torpedo can be deter- 
mined by the study of a model in a water tunnel or 
wind tunnel or by towing a full-scale body in a towing 
tank. Practically all torpedoes are statically unstable. 
This means that, if the torpedo is held at its center of 
mass while the water is pumped past it or if it is 
towed by an attachment at its center of mass, it will 
not continue to travel with its axis parallel to the 
direction of motion. A slight displacement will cause 
it to turn one way or another and set itself at a con- 
siderable angle. A bare torpedo body would probably 
set itself at nearly 90 degrees in the direction of mo- 
tion, but, because of the presence of the tail fins and 
the shroud ring, an actual torpedo will not turn this 
far. 

Although this type of static instability is very 
striking, it is of relatively little significance in con- 
nection with the running behavior of the torpedo. 
The reason for this is that, when the torpedo axis 
turns slightly one way or another, the direction of 
motion of a free torpedo driven by its own propellers 
also changes. It is possible to set up a criterion, as is 
done in Chapter 8, for what is called dynamic sta- 
bility. A body is dynamically stable if, when it is 
displaced from its straight course, it takes up another 
relatively straight-line course in a direction slightly 
different from the original. On the other hand, if a 
body is unstable dynamically and is displaced slightly 
from its course, it will go into a circle and continue to 
turn with a definite radius of curvature. A body that 
is dynamically unstable can be steered, but it imposes 
a very considerable load on the steering mechanism. 
A body that is dynamically stable can be steered 
with much less anticipation in the rudder correction. 
A body that is statically stable will need very little 
steering since static stability usually corresponds to 
a high degree of dynamic stability, but it will be 
very difficult to turn. A body that is dynamically 
stable will always turn in the direction corresponding 
to its rudder position, while if it is dynamically un- 
stable it may turn in the opposite direction for small 
rudder angles. 

Owing to the simple shapes of conventional torpedo 
bodies, it is possible to set up a scale of stability in 
terms of the hydrodynamic constants of the body. At 
the one end is a region of dynamic instability, and at 
the other end is a region of static stability. Between 
these is a region of static instability but of dynamic 
stability, in which a steering device can turn the tor- 
pedo in a reasonable circle and also keep it on a 


14 


OUTLINE AND SUMMARY 


straight course without too great limitations being 
imposed on the steering mechanism itself. 

The criterion for dynamic stability cannot be ex- 
pressed entirely in terms of coefficients that can be 
determined from straight-line motion in a to wing- 
tank or from ordinary measurements in a water 
tunnel. Additional measurements that can be made 
on a free-running body or that can be made by moy- 
ing the body in a large circle in a towing basin can 
giye these constants. 

This indicates the importance of a careful hydro- 
dynamic study of any torpedo body. Such a study 
must be made not only in straight-line motion but 
also in cuiwed motion. In addition, it is important 
that the study be made with propellers attached and 
possibly eyen power-driyen. The measured y allies of 
the hj^drodynamic constants appear to depend 
strongly on whether the propellers are present or 
absent and possibly on whether they are driyen or 
are free. 

3.3.2 Steering INIechanisms 

A steering mechanism may be either a two-position 
mechanism or a proportional mechanism. A propor- 
tional mechanism is such that the rudder displace- 
ment of the torpedo is proportional to the amount by 
which the torpedo axis departs from its prescribed 
direction. In a two-position mechanism the rudder is 
thrown hard oyer to one side or the other as soon as 
the torpedo departs more than a prescribed amount 
from its proper direction. There are also some other 
types that may be regarded as intermediate. 

A proportional mechanism may be such that, com- 
bined with the hydrodynamic properties of the body, 
it results in unstable oscillations of increasing magni- 
tude. This may be brought about if the control is too 
stiff, that is, if the rudder displacement diyided by 
the deyiation of the torpedo from the prescribed di- 
rection is too large a constant. Such an unstable 
system is, of course, unsatisfactory because the tor- 
pedo wanders widely from side to side. Instability 
of this kind can be corrected by reducing the amount 
of the rudder throw or by reducing the rudder area. 
Such a remedy reduces the curyature of the torpedo 
path and makes it more difficult to turn the torpedo 
in a prescribed direction. It also makes slower the 
correction of the course after a disturbance. In de- 
signing a steering mechanism, a proper balance must 
be struck between the necessity for stability and the 


necessity for a sensitiye control or for a quick restora- 
tion after a disturbance. 

Probably one of the most important sources of in- 
stability and unsatisfactory performance of the steer- 
ing mechanism is the time delay that exists between 
the motion of the torpedo and the motion of the 
rudder. In an ideal control, the rudder is displaced 
just as soon as the torpedo departs from its course, 
but in most practical deyices the rudder displacement 
lags a little bit behind the torpedo displacement. This 
lag may amount to one- or two-tenths of a second 
and is probably one of the principal reasons why some 
control mechanisms will work at low speeds and not 
at high speeds. The time delay that can be permitted 
is just inyersely proportional to the speed at which 
the torpedo is running so that the time delay that is 
troublesome at 30 knots may cause instability if the 
torpedo runs at 40 knots or more. 

The two-position control always results in oscilla- 
tion of the torpedo about its course. If, howeyer, this 
oscillation can be made of high enough frequency 
and of low enough amplitude, it is not serious. For 
example, if the torpedo oscillates at 1 c and turns 
through one-tenth of a degree in this time, the result 
will be quite insignificant. 

The time lag in a two-position control will reduce 
the frequency of oscillation and will correspondingly 
increase the amplitude. Hence, in this type of control 
also, the time delay must be limited to an amount 
that does not produce too much oscillation. In fact, 
it is probable that the limitation on the time delay is 
more seyere in the case of the two-position control 
than in the case of a proportional control. Neyer- 
theless, if a two-position control is properly designed 
and built, it will steer just as well as a proportional 
control. 

3.3.3 Depth-Control Mechanism 

The depth-control mechanism may operate in the 
same two ways as the steering mechanism. It may 
produce an eleyator deflection that is proportional to 
a giyen signal or combination of signals, or it may put 
the eleyator either hard up or hard down. 

The simplest form of depth mechanism might sup- 
posedly be a simple pressure bellows attached to the 
eleyator in such a way that the deflection of the 
eleyator is proportional to the amount by which the 
hydrodynamic pressure differs from its yalue at the 
desired depth. If, then, the torpedo were too deep, 


CONTROL OF UNDERWATER RUN 


15 


the pressure would be too great, and the elevator 
would turn up. It turns out, however, that such a 
mechanism is unstable and that the torpedo would 
oscillate widely about its set depth. 

In order to get adequate depth-keeping, it is essen- 
tial to have some kind of an anticipatory device. 
Thus, if the torpedo is running at its set depth and 
turns in order to start up, the depth mechanism must 
detect this initial change in pitch which takes place 
before the depth has changed at all. Similarly, when 
the torpedo is above its set depth but has turned so 
as to start down, the depth mechanism must recog- 
nize this fact and restore the elevator to its neutral 
or up position even before the torpedo has reached 
the proper depth. 

i\Iany kinds of anticipatory devices have been sug- 
gested. The one in most common use, and the one 
that appears the simplest, is a pendulum. This pen- 
dulum indicates the angle of inclination of the tor- 
pedo axis. In a proportional depth-control mechanism 
the position of the rudder is then made proportional 
to a combination of the departure from the set depth 
and the inclination. 

This kind of depth control works very satisfac- 
torily under suitable conditions. A suitably devised 
mechanism can control a torpedo, running at 45 
knots, to within 6 in. of its prescribed depth. Never- 
theless, the pendulum has numerous disadvantages. 
In the first place, the pendulum has a natural period 
of its own, and it is important that this natural period 
does not come too near to the natural period of depth 
oscillation of the torpedo. Furthermore, the torpedo 
can oscillate at a certain frequency at which the 
pendulum will not indicate any oscillation at all. This 
is called the frequency of antiresonance, and it de- 
pends on the location of the pendulum within the 
torpedo. These two points can be taken care of to a 
large extent in the design and location of the pendu- 
lum, but they do impose certain limitations upon the 
kind of pendulum that can be used. 

Perhaps the most serious objection to the pendu- 
lum is its response to acceleration. Particularly in the 
case of an aircraft torpedo, the deceleration of the 
torpedo on entering the water moves the pendulum 
forward and turns the elevator up. If the torpedo 
remains right side up, this may lead to excessive 
broaching. If the torpedo rolls over 90 degrees to one 
side or the other, this action of the pendulum may 


cause large hooks, whereas, if it turns over com- 
pletely, a deep dive may be the result. For the same 
reason a submarine torpedo tends to dive when 
ejected from the tube since it accelerates at that time, 
throwing the pendulum back and hence the elevator 
down. Furthermore, the pendulum is sensitive to 
changes in speed occasioned by the uneven supply of 
the fuel to the turbines. This may cause erratic depth- 
keeping. In spite of all these objections, however, no 
other mechanism has yet been extensively used, and 
it may well be that the pendulum, because of its sim- 
plicity, will always be the most satisfactory and that 
its disadvantages can be minimized by suitable de- 
sign. Future changes may emphasize some disad- 
vantages. 

One of the means suggested for eliminating the 
pendulum is to use in its place a device sensitive to 
the time rate of change of depth. This is not too 
satisfactory because the torpedo does not begin to 
climb until after it has changed its inclination. 
Nevertheless, the analysis shows that such a mecha- 
nism can be constructed to give stable depth-keeping 
if the constants are carefully selected. In particular, 
if a pendulum is also included, stability can be guar- 
anteed, and the disadvantages of using a pendulum 
alone can be minimized. 

Another suggestion has been the use of a gyroscope 
for indicating the vertical. The principal difficulty 
with this method is that a free gyroscope will not 
maintain its direction with sufficient accuracy for the 
depth-keeping. It is possible, however, to introduce a 
procession to keep the axis of the gyroscope perpen- 
dicular to the main position of the axis of the torpedo. 
Although this device has been studied, it has not yet 
been given extensive service trials. 

The principal result of the study of depth-keeping 
is the development of a theory by which it is possible 
to predict the performance of any projected mecha- 
nism. If a depth mechanism has been built, its char- 
acteristics can be examined in the laboratory on a 
tilt table or another similar device. If the device is 
only projected, its expected characteristics can be 
used to determine the behavior of the torpedo under 
its action. Tests have shown that the theory gives a 
close and fairly detailed description of torpedo be- 
havior so that there is no longer any excuse for the 
laborious production of depth mechanisms that can- 
not be expected to operate at all. 






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HYDRODYNAMICS AND AERODYNAMICS 

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Chapter 4 

HYDRODYNAMIC AND AERODYNAMIC 
FORCES AND MOMENTS 


A TORPEDO is fundamentally an underwater weapon 
and so it is natural, although not necessarily 
correct, to consider its form primarily from a hydro- 
dynamic point of view. The principal objectives have 
always been the minimization of drag and the 
achievement of a reasonable amount of stability 
within the limitations of overall dimensions imposed 
by the payload and the launcher. In an aircraft tor- 
pedo, however, the air travel and water entry must 
be given careful consideration as well. At the present 
time it appears that the water-entry phase of the 
motion is of dominant importance in determining the 
structure since less can be done to remedy the effects 
of poor form or inadequate strength in this connec- 
tion than in others. For, in principle at least, the air 
travel can be stabilized b}" additional members that 
are removed on water impact, the underwater tra- 
jectory can be improved by the control system, and 
any extra drag can be offset by an increase in power 
plant. It is, however, still of the greatest importance 
to understand the hydrodynamic forces and moments 
experienced by the torpedo since some freedom in 
shape design is still available after water-entry re- 
quirements are satisfied and, in any event, the design 
of the control system and the power plant depends on 
these factors. 


4.1 DRAG 

The drag is the component of hydrodynamic force 
in the direction of the instantaneous velocity vector 
and always opposes the velocity. Since it is balanced 
mainly by the thrust of the propellers, it is of domi- 
nant importance in determining the power required. 
For very small speeds the drag is entirely due to 
viscous forces in the region of laminar flow surround- 
ing the torpedo and hence is proportional to the 
speed. For speeds of practical interest the boundary 
layer is turbulent while the flow slightly farther out 
is smooth, and cavitation is usually not well enough 
developed to be significant. In this regime the drag 
tends to be more nearly proportional to the momen- 
tum transferred to the torpedo by turbulent masses 
of water in the boundary layer (dynamic pressure), 
that is, to the square of the speed. Experiments on 


models in the high-speed water tunnel, experiments 
on airships and airship models, and modern theories 
of the turbulent boundary layer indicate that the 
exponent is in the range 1.8 to 1.9. Thus the power 
required varies as the 2.8 to 2.9 power of the speed. 

The influence of the drag on the underwater tra- 
jectory at entry is of secondary importance and 
shows up principally in the deceleration that con- 
tinues after the collapse of the bubble. The effect of 
drag on the operation of the controls during the 
steady run is of still less importance. In both these 
cases it is a sufficiently good approximation to assume 
that the drag is proportional to the square of the 
speed, and this simplifies the analysis of Part III. 
Therefore, a drag coefficient can be defined as at 
the end of Section 4.3. 


4.2 MOMENT 

An elongated object such as a torpedo moving 
through a nonviscous fluid with a fixed acute angle 
of attack between its longitudinal axis and its ve- 
locity vector would be expected to experience an 
upsetting moment which would tend to increase the 
angle of attack. The moment observed experimen- 
tally in straight-line motion agrees with this predic- 
tion in sign, in the near proportionality to attack 
angle for moderate angles, and in the proportionality 
to the square of the speed. Since this moment, as well 
as the remainder of the hydrodynamic moments and 
forces, is made up of the totality of normal forces or 
pressures exerted on various parts of the hull and 
empennage, and since these pressures are propor- 
tional to the dynamic pressure (p '2)F“ for fluid 
density p and speed F, it is convenient to define a 
dimensionless moment coefficient Cm by the relation 


il/ = moment = 



VKMCm , 



where A and I are the largest cross-sectional area and 
the overall length of the torpedo. This coefficient is 
found to be practically independent of speed. For a 
symmetrical body moving along its axis Cm = 0; its 
derivative with respect to attack angle is positive, 


19 


20 


HYDRODYNAMIC AND AERODYNAMIC FORCES AND MOMENTS 


meaning that the moment tends to increase the angle 
of attack. 

The addition of tail fins or a shroud ring to the aft 
end of a bare torpedo hull always decreases the mag- 
nitude of the derivative of Cm with respect to attack 
angle and could, in principle, make it negative if car- 
ried far enough. Actual torpedoes appear always to 
have a positive derivative in the neighborhood of zero 
angle of attack. In a similar way the position of the 
control surfaces affects the value of Cm- When these 
are mounted on the ends of the fins, deflection of the 
control surface in one direction produces a force on 
them in the opposite direction and hence a moment 
that tends to swing the nose of the torpedo in the 
direction of the deflection. 


4.3 CROSS FORCE 

Although an object moving through a nonviscous 
fluid with constant velocity and orientation expe- 
riences no force perpendicular to its velocity vector, 
the actual torpedo does. This is due to the effect of 
viscosity in unbalancing the dynamic pressures on 
various parts of the body. The resultant lateral force 
L can be expressed in terms of a dimensionless force 
coefficient C l 

L = V^ACl . (2) 


For a symmetrical body moving along its axis 
C L = 0) its derivative with respect to attack angle is 
positive, meaning that an attack angle produced by 
deflection of the nose in one direction will result in a 
force in that direction. This agrees in sign with the 
well-known lift force observed in airships flying with 
nose up. From the discussion of Section 4.2 it follows 
that an increase in empennage area increases the 
magnitude of Cl and that a deflection of a control 
surface in one direction produces an increment of 
force in the opposite direction. 

By analogy with the treatment of the lateral force, 
it is convenient to represent the drag force discussed 
in Section 4.1 in terms of a drag coefficient CT: 


1) = drag = 



V‘^ACd 



Although Cl) is less independent of speed than Cm or 
Cl, it is often a useful approximation to regard it as 
constant with respect to changes of speed. Experi- 
mental evidence indicates that Cl* is also independent 
of the attack angle and the deflection of the control 
surfaces for moderate values of these quantities. 


4.4 DAMPING MOMENT AND 

FORCE 

The discussion thus far in this chapter is limited to 
the forces and moments encountered in straight line 
motion. When the center of gravity [CG] of the tor- 
pedo moves in a curved path, it is convenient to 
divide the net forces and moments into two parts. 
The first is that experienced by the torpedo when 
moving along a straight line, with the attack angle 
that the CG has in the curvilinear motion. The 
second part is defined to be the remainder and is re- 
ferred to here as the damping force or moment. 

It is apparent that, in general, curvilinear motion 
of the CG results in rotation of the torpedo about a 
transverse axis that is perpendicular to the plane of 
the motion. This rotation combined with the for- 
ward motion results in an attack angle that varies 
along the length of the torpedo and hence a change in 
the pressure distribution referred to in Section 4.2. 
The damping force and moment would therefore be 
expected to be proportional to the angular velocity oo 
of the torpedo, at least for small w, since the pressure 
at each point is proportional to (p 2)F“ and the 
change in attack angle at each point is approximately 
cox/T^, where x is the distance from the CG. The 
resultant force would be expected to be nearly per- 
pendicular to the axis of the torpedo so this compo- 
nent is called Fco and the drag component is neg- 
lected. Similarly, the resultant moment will have the 
form /voj. These can then be written in terms of di- 
mensionless coefficients C f and Ck that are expected 
to be independent of speed : 



The damping moment always opposes the angular 
velocity, and, in general, a rotation that moves the 
nose in one direction gives rise to a lateral force in 
the same direction. 

Although there is some evidence that Cf and Ck 
depend on attack angle for other bodies, the data on 
torpedoes is so meager that it is customary to regard 
them as constants. The discussion of Section 4.2 
indicates that the magnitudes of both of these co- 
efficients should increase with increase in empennage 
area. 

Torpedo propellers generally add a cross force at 
the torpedo tail and hence influence all the hydro- 
dynamic coefficients. This will be discussed further. 


Chapter 5 

AIR FLIGHT OF A TORPEDO 


5.1 THEORY OF THE AIR FLIGHT 

5.1.1 The Equations of Motion 

I X CHAPTER 4 the forces acting on a torpedo during 
its motion through either air or water were de- 
scribed, and dimensionless quantities were defined in 
terms of which it is convenient to express them. In 
Chapter 7 the equations of motion of the torpedo in 
water will be more fully discussed in terms of these 
coefficients. Here the approximate equations will be 
written down and solved to the extent necessarv for 

K/ 

an understanding of the air flight. 

Let the origin of coordinates be at the position of 
the torpedo CG at the time of release, and let the 
X axis be horizontal in the direction of forward mo- 
tion. Let the y axis be vertically downward, and let 
the z axis be perpendicular to these two so as to make 
a right-hand system of coordinates. 

If the effects of the air are to be neglected, the 
equations of motion are very simple and can be 
written directly as 

x = 0 , 

y = a, ( 1 ) 

z = 0 . 

The solution of these equations gives the well-known 
parabolic trajectory. 

If the resistance of the air is to be taken into 
account, the eciuations become a little more compli- 
cated for the drag is proportional to the scpiare of the 
velocity and always acts opposite to the direction of 
motion. This leads to the equations 

X = —kxV , 

y = g-W\ ( 2 ) 

z = —kzV, 

where k = CdpAJ2M. Cd is the drag coefficient de- 
fined in Chapter 4, p is the density of the air,"^ A is 
the cross-sectional area of the torpedo, and M is the 
effective mass of the torpedo. The drag coefficient 

^ The change in density of the air with altitude is neglected, 
and the drag coefficient is constant over a sufficient range of 
yaw and pitch angles. 


Cd is that pertaining to the torpedo and its stabiliz- 
ing appendages. These appendages maj^ give rise to 
the major part of the air resistance. 

The solutions of equations (2) will be discussed in 
the Section 5.2 where it will be shown that, because of 
the air resistance, the horizontal velocity will tend to 
decrease during the drop. At the same time the ver- 
tical velocity will be less than it would be at the cor- 
responding time if the drop were in vacuum. This can 
be described roughly by saying that the effective 
acceleration of gravity is somewhat reduced by the 
air resistance. It will also be shown that a torpedo 
dropped from a given height will have a reduced 
range and an increased time of drop compared with a 
corresponding drop in vacuum. 

In addition, because of the air resistance, the ver- 
tical and horizontal motions will no longer be inde- 
pendent. For example, the time of drop for a torpedo 
launched horizontally from a given altitude will de- 
pend slightly on the release velocity. 

It was shown in Chapter 4 that, in addition to the 
drag force due to the air, there are transverse forces. 
These are zero when the torpedo is traveling in a 
straight line parallel to its axis but come into play 
when the torpedo axis deviates from the direction of 
motion or when the direction of travel is changing. 
These forces will be neglected at first in the treatment 
given here because they are small and because the 
torpedo oscillates in its flight so that the forces act 
alternately in one direction and then in the other. 
For these reasons it appears that their neglect is 
usually justified in any practical treatment of the air 
trajectory. 

In addition to the motion of the center of mass, it 
is necessary to study the motion of the torpedo about 
this center because the attitude of the torpedo as it 
enters the water is of the utmost importance. Let 6 
be the angle between the tangent to the trajectory 
and the x-z plane, and let 6 be positive when the tor- 
pedo is falling. Then tan 6 = y x when z = 0. In 
addition, let a be the angle in a vertical plane between 
the torpedo axis and the trajectory, and let it be 
positive when the torpedo is nose-up. Similarly, let 
\p be the angle in a horizontal plane between the tor- 
pedo axis and the trajectory. These two angles are 


21 


22 


AIR FLIGHT OF A TORPEDO 


small enough to be treated entirely independently. 
The equations of motion are then 


a ya qa = pd -\- B , 
^ = 0 . 



The first equation contains no term in B because the 
restoring force is such as to urge the torpedo to lie 
parallel to the trajectory. The terms in B and B appear 
because the direction of the trajectory is changing. 
The second equation has zero on the right side since, 
to the accuracy considered here, the projection of the 
trajectory on the horizontal plane is a straight line. 

The coefficients q and q' are expressible in terms of 
the moment coefficient Cm defined in Chapter 4, 


In the first place, the axes were so chosen that z is 
zero at the time of release. It will then be assumed 
that i never becomes comparable with x so that in 
equations (2) V will be taken as (x“ + ^“) The equa- 
tions to be treated will then be 

X = —kx{x- -f y-y^ , 

ij = g -ky(x- -f (6) 

z = 0 . 

The procedure will be to first get a suitable ap- 
proximate solution to equations (6). This will then 
provide the value of 0 as a function of the time to 
insert in equations (3) . 

The initial conditions to be used at the time t = 0 
are as follows. 





In this expression Cm refers to the moment around a 
horizontal transverse axis through the center of mass, 
and I is the effective moment of inertia about this 
axis, q' can be expressed in a similar fashion in terms 
of the moment around the vertical axis. With some 
types of stabilizers these two moments are quite 
different, but the purpose of the stabilizers is to make 
them both positive. With air stabilizers making 
5 > 0 and q' > 0 the torpedo, during the air flight, 
is statically stable, thus insuring a high degree of 
dynamic stability. Because of this high degree of 
dynamic stability and since the air disturbances are 
quite small, a torpedo in air does not have to be 
steered. 

Similarly, referring to Chapter 4, 

p = CK-Af-. (5) 

2 1 

The solutions of equations (3) will be discussed and 
the nature of the motion illustrated in later sections. 


5.1.2 Simplification for Solution 

Although only parts of the complete ecpiations are 
written down in the previous section, they are still 
too complicated to make a complete and rigorous so- 
lution profitable. Therefore, certain additional sim- 
plifying assumptions will be made to aid in under- 
standing the nature of the motion. 


^(0) = 2/(0) = ^(0) = 0 because of the location of 
the origin, 

±0 = horizontal component of release velocity, 

7jo = vertical component of release velocity, 

io = 0, 

ao = pitch angle of torpedo at release, 

do = pitching angular velocity of torpedo at re- 
lease, 

\po = yaw angle of torpedo at release, 

\po = yawing angular velocity at release. 

5.1.3 Solution for the Trajectory 

Even the simplified equations (6) cannot be solved 
exactly but must be given an approximate treatment. 
Hence define a quantity u as 

u = tan B = - . (7) 

X 


Then from equation (6) it follows that 


and 





u = gk(l + . 


(8) 

( 9 ) 


This last differential equation in u can be solved in a 
power series, so let 


^ . ( 10 ) 

7i =0 


THEORY OF THE AIR FLIGHT 


23 


From the differential equation it follows that 
Co = Uo = tan do , 

9 

Cl = Uo = — , 

i'o 


It is sometimes necessary to know the total ve- 
locity as a function of the time. The velocity at 
water entry is one of the initial conditions of the 
underwater trajectory, and the velocity in the air 
determines the forces acting on the torpedo. Since 
the velocity in the 2 direction is neglected, 


C 2 


Ho 


2 


9^>^ 

— sec do , 

2 


r = (i-2 + = i(i + = JL _ (15^ 

ku 



g-k . ^ 

- — sm ^ 0 , 
6i-o 


If the terms containing are retained in both the 
numerator and the denominator, this becomes 



= {g cos^ do + kxo~ tan do) . 


2c 2 “h I2 c 4^^ 
A^(ci 4“ 2 c2^ T Sc^t~) 


(16) 


Each quantity desired can then be expanded in 
powers of t. 


u = tan d = Uo — sec do t~ 


( 11 ) 


Xo 


From (8) it follows that 


X = 


9 

u 


= Xo 


so that 


ok 

1 — kxo sec dot + (A:“Xo“ sec- do — — sin do) f 

2 


(12) 


The velocity first falls off because of the air resis- 
tance and then later increases because of the accelera- 
tion of gravity. The initial decrease is not present 
when k = 0, and the later increase is less rapid than 
in the vacuum trajectory because of the air resis- 
tance. The value of t corresponding to a trajectory 
angle d can be obtained from equation (11). 

For some purposes, in particular for estimating 
the stabilizing moments on the torpedo during its 
air flight, it is convenient to use an average value of 
the velocity, 1^. For torpedo launchings it is often 
desired to produce a specified entry angle, and to 
know the time average of V up to the time this angle 
is attained. With sufficient accuracy the time over 
which the average is desired is given by 



xdt = Xot — 


kxo^ 

sec dot- 

2 


+ ^ {kHo^ sec^ do 


9kxo 

2 


sin do)d‘ • • • 


(13) 


Equations (11) and (12) show that the horizontal 
velocity is always less than the initial velocity and 
the horizontal distance traveled in a given time is 
less than would be traveled in vacuum. 

Similarly, from equation (7), 


y = ux = g 


Co T Cit -j" C2^^ 


Cl 2 c2^ 


Vo + 9t + 


9kxo 


sec dot- 


ok 

1 -f- kxo sec dot + — sin dot~ 

2 


(14) 



9 /ci + 2C2^e + 3C34““l“4C4^« 

kxoitsinde — tan^o) ^ V 


Cl 


To the accuracy of this expression F, the average 
velocity during the drop is a function only of the 
horizontal component of the release velocity, the 
trajectory angle at release, and the trajectory angle 
at entry. For many purposes, this may be treated as a 
constant velocity during the drop. 

In the treatment thus far the air force has been 
considered as acting along the trajectory and through 


24 


AIR FLIGHT OF A TORPEDO 


the center of mass. In other words, drag only has been 
considered. As a matter of fact, other forces are also 
acting. Among these the most important in its in- 
fluence on the trajectoiy is the lift force. This acts 
perpendicular to the direction of motion and is pro- 
portional to the pitch angle of the torpedo. Since the 
torpedo oscillates during its flight so as to have alter- 
nately nose-up and nose-down pitch, this effect will 
be relatively small. There will be a small residual 
effect due to the fact that the torpedo oscillates about 
a position that is slightly nose-up because of the 
curvature of the trajectory. An additional mean 
pitch, either up or down, can be produced if the 
stabilizer is not s^^mmetrical. 

A significant effect on the trajectory may also be 
produced if the stabilizers are not quite symmetrical. 
A stabilizer, either intentionally or unintentionally, 
can be essentially an airplane wing and carry the 
torpedo much farther than it would travel without 
such an attachment. In case a stabilizer is designed 
to perform this function, provision must also be 
made to stabilize against roll. This has been done in 
the British air stabilizer, but it requires a consider- 
able complication of the equipment. 

5.1.4 Pitching and Yawing Motions 

Pitch Oscillations 

Thus far the trajectory of the torpedo has been 
treated b}^ neglecting the effect of pitching and yaw- 
ing motions. These motions have been neglected 
since their net effect is small because of their oscilla- 
tory nature. An expression for the resultant velocit^A" 
has been obtained and also a relation derived for the 
average velocity during the drop I" so that for any 
drop V may be regarded as constant and equal 
to y. 

Since in the equations (3) for the pitching and 
yawing motions p and p' are proportional to T^, and 
q and q' are proportional to we can regard these 
quantities as constant during the drop, and their 
magnitude will be given by the constant V which 
depends on the horizontal release velocity" and tra- 
jectory angles at release and entry. 

It is well known that if a torpedo is released in air 
without any appendages, such as stabilizers, it will 
tumble. It is necessary to install a stabilizer in order 
to create a moment that tends to line the torpedo up 
vith its trajectory. With such a restoring moment, the 
torpedo, in general, will oscillate about its mean pitch 


angle, and it is also necessaiy to have these oscilla- 
tions damp out as rapidW as possible. In the Mark 13 
torpedo this damping is more than doubled by means 
of the drag ring, or pickle barrel. 

If the axis of the torpedo makes an angle with the 
trajectory at release (the pitch angle at release) or if 
the axis of the torpedo is rotating with an angular 
velocity at release (the pitching angular velocity at 
release), the torpedo will proceed to oscillate while 
it is falling. 

However, even if the pitch angle and pitching 
angular velocity at release are both zero, the torpedo 
will still oscillate during its fall. This is due to the 
fact that the center of gravity of the torpedo is mov- 
ing along a curved trajectory and the axis of the tor- 
pedo at release may be along the trajectory, but it 
must turn to keep up with the changing direction of 
travel. 

For the pitch, the first of equations (19) must be 
solved. 

d pa qa = pd -\-d . (18) 

The right-hand side of this vdll be treated as a 
known function of time since the approximate equa- 
tions for the trajectoiy have already been solved. 
This is then an inhomogeneous differential equation, 
and its complete solution is obtained from the sum of 
the solution of the homogeneous equation 

d + pd qa = 0 (19) 

and a particular solution of the inhomogeneous equa- 
tion. The solution of the homogeneous equation will 
be chosen so as to depend on the release conditions of 
the torpedo, the pitch angle, and pitching angular 
velocity at release. The particular solution will then 
be chosen to be independent of these initial condi- 
tions and will then represent the value the pitch 
angle a would have if both it and the pitching angu- 
lar velocity at release were zero. The particular 
solution is then the part of a due to the fact that the 
torpedo traverses a curved trajectory. 

The solution of the homogeneous equation is 

a,{t) = {ao COS 0,1 + sin , (20) 


where co = V ^ — (p/2)-, 

ao = pitch angle at release, 

do = pitching angular velocity at release. 


25 




DISCUSSION AND ILLUSTRATION OF THE THEORY 


It should be noted that : q is proportional to and 
p is proportional to so that co is proportional to T". 

The particular solution of equation (18) leads to 
integrals which cannot be simply expressed. How- 
ever, by partial integrations, a solution is obtained 
which is a very good approximation and is certainly 
sufficiently accurate for the problem being treated. 
Thus the particular solution of equation (18) may 
be taken as 

^2(0 — — [^(0 V + ^(0) sin (co^ — r?) ] (21) 


4/{t) = cos co't + ^ sin co'^y (24) 

where co' = 's/ q' — (p'/2)“ , 

= yaw angle at release, 
xpo = yawing angular velocity at release. 

The solution is that of a damped harmonic mo- 
tion. In this case, if the torpedo were released with 
zero yaw angle and yawing angular velocity (xpo = 0 
and xpo = 0), the yaw angle would always be zero 
during the drop. 


where r = q = A/co-f (p 2)- and is proportional 
to T, 

77 = tan“^( — 2co p) and, since co and p are both 
proportional to T for a given torpedo with given air 
appendages, r? is a constant. 

Equation (10) gives tan ^ as a function of the time 
and from it the value of B{t) and 0(0) can be obtained. 
Combining the two integrals gives the complete so- 
lution for the pitching moment. 


a{f} = <; - 

.CO 


p • 

coq;o — - 0(0) sin rj 


cos co^ 


+ 1 


CO 


cio + p — + - 0(0) cos 7] 


sin cot 


-f — 0(0 sin 7] . (22) 

cor 

Examination of the pitching motion given by this 
solution indicates that the motion may be described 
as damped oscillations about a mean position which 
is itself decreasing with time. The first term repre- 
sents damped oscillations, while the second term is 
not oscillatory but decreases monotonically with 
time. It is clear that increasing p, which may be 
achieved by increasing the release velocity ± 0 , de- 
creases the magnitude of the pitch angle. This is even 
more effective than increasing the height of drop ye 
since p oc XQ{ye)^\ 

Yaw Oscillations 

We may now consider the yawing motion during 
the air flight. This is described by 

^ + p'i^ + q'"^ 0 . (23) 

The solution of this equation yields the yaw angle 
as a function of time, namely : 


5.1.5 Further Approximations 

In this section the equations of motion were sim- 
plified and a solution obtained in which the forces 
acting on the torpedo were its weight and the drag 
forces. With these forces the trajectory was obtained 
and is given by equations (13) and (14). This tra- 
jectory lies in one plane since the forces perpendicu- 
lar to the initial plane of motion are negligible. With 
these forces solutions to the pitching and yawing 
motion were obtained and are given by equations 
(22) and (24) . The pitching motion may be described 
as damped oscillations of the axis of the torpedo 
about a mean position which is not zero but is slightly 
nose-up and decreases with time, while the yawing 
motion is that of damped oscillations about a mean 
position of zero yaw. 

These results may be considered a first approxima- 
tion. For, having obtained the solutions for a(t) and 
4'it), one may apply a method of successive approxi- 
mations. Thus, substituting the expressions for a{t) 
and in more complete equations, the equations 
of motion of the trajectory may be solved again. The 
primary effect will be that the center of gravity of 
the torpedo will oscillate with very small amplitude 
about the trajectory. Then from this solution one 
may obtain an expression for d{t) and Using 

these expressions, a further approximation may be 
obtained for a{t) and 

These further approximations are not carried out 
here because the accuracy with which the various 
quantities are needed does not require them. 


5.2 DISCUSSION AND ILLUSTRATION 

OF THE THEORY 

Throughout Section 5.2, where illustrations and 
magnitudes are given, except when stated otherwise, 
they will always be given for the Mark 13 torpedo 


26 


AIR FLIGHT OF A TORPEDO 


with shroud ring, fitted during the air flight with the 
Mark 2-1 stabilizer and Mark 1 drag ring (pickle 
barrel). The constants are given in Section 5.4. 

5.2.1 Air Trajectory 

Plot of Horizontal Distance as a Function 
OF Time x{t) 

Figure 1 is a plot of x{t) for the special case of 
horizontal release, Oq = 0, as given by equation (13) 


Vertical Fall as a Function of Time y{t) 

Figure 2 is a plot of y{t) for the special case of 
horizontal release, Bq = 0, as given by the integral of 
equation (14). Here again the graphs are drawn for 
release velocities of 200 knots and 350 knots, and, for 
comparison, the vacuum vertical falls, given by 
y = gf^/2, are also drawn. 

From this figure it is seen that in a given time the 
torpedo in a vacuum will fall through a greater ver- 
tical distance than a torpedo in air. Or, equivalently, 


9000 


8000 


7000 


6000 


- 5000 


UJ 

o 

z 

< 

a. 

4000 


z 

0 

1 3000 
z 


2000 


1000 









VACUUM TRAJECTORY — ^0 . 

AIR TRAJECTORY— Xo ‘ 

VACUUM TRAJECTORY 

AIR TRAJECTORY— 200 KNOTS 

k = 4.5xi0‘* ft'' 



/ / 

/ 

/ 





/ / 

' / 

/ 

/ 





/ / 

/ / 

/ 

/ 





x*Lt 

1 

l//’’ 

// 





A 

//\ 

> l 


* x,i 

Y _ k ioT* 

2 

^ k*ko*t® 

3 


- ~ 

/ 

y> 

5?^ 













3000 


2000 


o 

z 

< 

cc. 


z 

o 

N 


1000 o 

X 


0 2 4 6 8 10 12 14 

TIME IN SECONDS 

Figure 1. Horizontal range versus time from release. 


in Section 5.1. From this figure one may readily see 
the effect of release velocity and drag coefficient 
(proportional to k) on the horizontal travel of the 
torpedo. Thus x{t) is plotted for Xq = 200 knots and 
350 knots, and at each of these release velocities the 
horizontal travel in a vacuum is plotted (k = 0), 
namely, x = Xot. 

From this figure it is readily seen that the hori- 
zontal range covered in any time is considerably less 
than the range covered in the corresponding time 
along a vacuum trajectory. 

Also from this figure it is noticed that increasing 
the release velocity also increases the amount by which 
the actual range (with drag) is less than the vacuum 
range. This is exactly as expected since the drag forces 
are proportional to V^. 



when both torpedoes are launched from the same 
altitude, the one in air will take a longer time to 
strike the water than the one in a vacuum. 

This effect is also seen to be larger the greater the 
release velocity. Here again this is true since the 
coefficient of the f term which represents a retarda- 
tion of the fall, increases roughly as Xq. 

Trajectory of the Torpedo y versus x 

From the preceding graphs one may easily plot y 
versus x, which is the actual path of the center of 
gravity of the torpedo during the air flight as shown 
in Figure 3. This figure shows that, for two torpedoes 
launched from the same height, the one traveling in a 
vacuum (without air forces) would travel farther 
than the one launched in air, and the amount by 
which it would travel farther increases with the ve- 




DISCUSSION AND ILLUSTRATION OF THE THEORY 


27 


locit 3 ^ of release because retarding forces increase as 
the square of the velocity. 

Horizontal Velocity as a Function of Time x(t) 

In Figure 4 x(t) is plotted for horizontal release at 
velocities of 200 knots and 350 knots. For compari- 



Figure 3. Distance of fall versus horizontal range. 


600 


550 


o 

o 

bJ 

V> 

IT 

hJ 

a 


UJ 

hJ 


450 


o 

o 

_) 


_) 

< 

t- 

z 

0 

tsf 

1 

0 

1 


400 


350 


300 









7 

• 

/ 

• 

/ 

s. 

• 


X = Xo 





• 

> 

• 

N 


X= Xo'l' 

2 2 
io t + k X 

3 2 

0 _ 




• 


• 

• 


VACUUM TRAJECTORY . 

AIR TRAJECTORY Xo= 350 KNOTS 

VACUUM TRAJECTORY . 


- 

A 

R TRAJE( 

k = 

:tory-- 

4.5 X 10'® 

_ ^0 * 

FT-‘ 

KNOTS 





X = Xo 


X^Xo-kx 

oh + K^Xo 



5 2 
t 




350 


300 


»- 

o 

z 

z 

> 

O 

o 

UJ 

> 


250 < 


o 

N 

a; 

0 

1 


200 


6 8 
TIME IN SECONDS 


10 


12 


14 


Figure 4. Horizontal velocity versus time from re- 
lease. 


son, the vacuum horizontal velocity is drawn, this 
being a horizontal straight line on the graph with 
magnitude equal to the release velocity Xq. It is seen 
that the horizontal velocity decreases with time in- 
stead of remaining constant, and the rate at which 
it decreases is approximately proportional to Xq^. 


Vertical Velocity versus Time y(t) 

From Figure 5 it is seen that the vertical velocity 
with air forces acting on the torpedo is less than the 
vertical velocity without the air forces. For a vacuum 
trajectory the graph, y(t) is a straight line with a 
slope equal to g. The drag force serves to decrease the 



</> 

H 

O 

z 


o 

o 

Ui 

> 

< 

o 


q: 

UJ 

> 


Figure 5. Vertical velocity versus time from release. 


slope with increasing time. Increasing the release 
velocity causes the vertical velocity to deviate more 
from the vacuum trajectory value. 

It is interesting to note the terminal velocity of the 
Mark 13 torpedo. This velocity is defined as the ve- 
locity the torpedo would have if it fell for an infinite 
time or, alternatively, the velocity the torpedo has 
when the drag forces are equal and opposite to the 
weight. This is readily seen to be terminal velocity 
= g Ik = 848 ft per sec, and without the Mark 1 
drag ring = 1,325 ft per sec. 

Resultant Velocity 

In Figure 6 the resultant velocity V = -\/ x' -f- y^ as 
given by equation (16) is plotted for horizontal re- 
lease, do = 0. Curves are shown for a vacuum tra- 
jectory and for the trajectory with air forces acting 


28 


AIR FLIGHT OF A TORPEDO 


on the torpedo, with release velocities of 200 knots 
and 350 knots. For horizontal release, one should 
expect V at first to decrease and be less than Xo, 
whereas for larger values of t after the acceleration 
due to gravity has become important, V should be 
larger than i’o- For a vacuum trajectory T" ^ jo during 
the entire drop. 



cn 

h" 

O 

z 


o 

o 

_j 

u 

> 


< 

I- 

o 


Figure 6. Resultant velocity versus time from re- 
lease. Also average velocity in time corresponding to 
de = 23°. 


From Figure 6 it is seen that the resultant velocity 
curve is of this form. At the higher release velocity of 
350 knots the resultant velocity is less than the re- 
lease velocity for times of drop of interest in most 
torpedo launchings. For the lower release velocity 
the resultant velocity is about equal to or larger than 
the release velocity for the times of interest. Thus, 
for example, if the torpedo enters the water with a 
trajectory angle de of aliout 21 degrees, with a re- 
sultant velocity of about 317 knots, it would cor- 
respond to releasing the torpedo with a horizontal 
velocity of 350 knots from an altitude of aliout 030 ft. 
The entry velocity in this case is about 33 knots less 
than the release velocity, even though the torpedo 
was 1 ‘eleased from 030 ft. 


In addition, in Figure 0 the value of for de = 23 
degrees as obtained from equation (17) is noted. 
From the form oiV{t) it is seen that, for the range of 
times of interest, one should be able to approximate 
V(t) hy some average value which is constant during 
the drop. For the two release velocities in question 
it is seen that in the range of time of interest is 
always close to F(0- 

In Figure 7, for future use, V in knots is plotted 
against x’o in knots for horizontal release with de = 23 


cn 

H 

O 

z 

is: 


l> 

> 

H 

O 

O 


liJ 

> 


u 

o 

< 

(T 

liJ 



150 200 250 300 350 400 450 

HORIZONTAL RELEASE VELOCITY, Xq IN KNOTS 


Figure 7. Average resultant velocity versus horizon- 
tal release velocity for de = 23°. 


degrees. It is noticed that for low velocities V > Xq, 
while for the higher release velocities F < Xq. 

Trajectory Angle d{t) 

For horizontal release at velocities of 200 knots and 
350 knots, d{t), as given by equation (11) and for a 
vacuum trajectory, is plotted in Figure 8. d(t) would 
be the slope of the curves in Figure 3 as a function of 
time. From this plot it is seen that the trajectory 
angle with the air forces acting is greater than the 
vacuum value at any time during the drop. The 
difference is larger the greater the release velocity 
and is, in general, not negligible. 

By means of graphs of x{t), x{t), y(x),V{t) 

and d{t) for horizontal release, with velocities of 200 
knots and 350 knots, with a k value corresponding 
to the Alark 13 torpedo fitted with the Mark 2-1 
staljilizer and Mark 1 drag ring, and the correspond- 
ing graph for the vacuum trajectory, the theory de- 


DISCUSSION AND ILLUSTRATION OF THE THEORY 


29 



TIME IN SECONDS 

Figure 8. Trajectory angle versus time from release. 


HORIZONTAL RELEASE VELOCITY IN KNOTS 
X = HORIZONTAL RANGE IN YARDS 
y = HEIGHT OF DROP IN FEET 


e = TRAJECTORY ANGLE IN DEGREES 
t = TIME FROM RELEASE IN SECONDS 
X HORIZONTAL VELOCITY IN KNOTS 



PiGURE 9. Nomogram for air flight with horizontal release of Mark 13 torpedo fitted with Mark 2-1 stabilizer and 
Mark 1 drag ring (pickle barrel). 


30 


AIR FLIGHT OF A TORPEDO 


veloped in Section 5.1 has been illustrated. In general, 
the nature of these curves may be explained by the 
fact that the external forces acting on the torpedo are 
proportional toT"^ and consequently will vary like 
the square of the release velocity. Thus the amount 
by which the actual trajectory differs from the 
vacuum trajectory increases roughly as Xq-. 

In general, the primary effect of the air forces is a 
deceleration so that from a given altitude it takes 
longer for a torpedo to reach the water when launched 
in air than when launched in a vacuum. The vertical 
velocity and the horizontal velocity and range are less 
than in a vacuum, and the vertical velocity is usu- 
ally less than the release velocity. Also, for the same 
launching condition, the torpedo will enter the water 
with a steeper trajectory angle when launched in air 
than when launched in a vacuum. 

The results can also be combined into a nomogram 
as shown in Figure 9. From release velocity and alti- 
tude, the entry angle and horizontal range, for hori- 
zontal release, can be determined quickly. 

5.2.2 Pitch Oscillations 

When the torpedo axis is nose-up relative to the 
trajectory (flat pitch), a is the pitch angle and is 
positive. When the torpedo is nose-down (steep 
pitch) the pitch angle is negative. 

Complete Solution of Pitch Oscillations 

Figure 10 is a graph of the pitching motion with 
the initial conditions of ao = — 4° and do = 15 degrees 
per sec, as given by equation (22) for horizontal 
release with velocities of 200 knots and 350 knots. 
Using the constants corresponding to the value of V 
associated with this Xo we have 

1. Xo = 338 ft per sec, U (knots) = 200 knots, 
w = 2.76, p = 0.588, 


quency and the damping so that the amplitude of the 
oscillation is considerably smaller. 

Particular Solution 

From Figure 10 it can be seen that the oscillations 
are not symmetrical about zero pitch angle but that 
there is a bias to a nose-up angle. This bias is repre- 



Figure 10. Pitch angle versus time from release 
oco = — 4°;do = 15 degrees per sec. 

sented by the particular solution, a 2 {t), of the equa- 
tion of motion for the pitch oscillations. The par- 
ticular solution is independent of ao and do, but it 
does depend on the release velocity. In Figure 1 1 the 



01 23456789 10 

TIME IN SECONDS 


2. Xo = 592 ft per sec, V = 330 knots, 

CO = 4. 56, p = 0.97. 

From this figure it can be seen that increasing the 
release velocity increases the frequency of the motion 
and, for the same initial conditions, decreases the 
amplitude, co which is the angular frequency of the 
motion is proportional to V which in turn varies like 
Xo, and p which is a measure of the damping is also 
proportional to V. Increasing V increases the fre- 


Figure 11 . Pitch angle given by particular solution 
versus time from release ao = 0°; do = 0 degree per sec. 

particular solutions are plotted for the two cases 
illustrated in Figure 10, Xo = 200 knots and 350 knots. 
It can be seen from this figure that for many applica- 
tions, the magnitude of the particular solution is not 
negligible. For small values of time the particular 
solution is rarely more than 15 per cent of the com- 
plete solution, but for large values of time it is usually 
more than 25 per cent. In general, it may be said 




DISCUSSION AND ILLUSTRATION OF THE THEORY 


31 


that, as release velocity increases, the particular solu- 
tion becomes more negligible compared to the com- 
plete solution. 

From Figure 11 it is evident that, as the release 
velocity increases, the magnitude of the particular 
solution decreases. The reason for this is that p, co, 
and r are all proportional to V, and 6 depends on the 
factor Cl = g Xo so that as a result a^it) is roughly 
proportional to 1/xo". 


the expression (25) for a(t). As t >- 00 ^ ci{t) ► 0, or 

eventually the mean pitch angle approaches zero. 
This occurs within the limit in which the torpedo is 
falling straight down. 

In Figure 12 a{t) as given by (25) is plotted for 
horizontal release with the same velocities as Figures 
10 and 11, 200 knots and 350 knots. To illustrate the 
behavior of 5(0, this figure has been plotted for time 
running up to 30 sec, although up to the present the 



0 5 10 15 20 25 30 

TIME IN SECONDS 


Figure 12. Mean pitch angle versus time from release. 


Equation (22) represents damped oscillations 
about a mean pitch angle that is itself decreasing 
with time. The mean pitch angle is given by 

. ( 25 ) 

T“Xq 1 (Co 4“ Cit “b 

This, of course, is also independent of ao and 5o. This 
mean pitch decreases relatively slowly with time 
because of the factor 1 + kxQ sec dot /[I + (co + Cit 
+ C 2 fy], while the other factors in equation (22) 
decrease rapidly with time since they are multiplied 
by the exponential ^ increases, the oscilla- 

tions about this mean pitch angle damp out, and the 
pitch angle approaches the mean position given by 


region of interest in torpedo launchings is up to about 
10 sec for 350 knots and about 6 sec for the 200-knot 
release velocity. For small times the amplitude de- 
creases with velocity; in fact, as has been shown, it 
decreases approximately like However, for large 
values of t this is no longer true since the terms in- 
volving t become much more important. 

It is important in connection with the behavior at 
water entry to note that the mean pitch angle of the 
torpedo is slightly nose-up. This may be explained 
physically by the fact that the center of gravity of 
the torpedo is traveling along a curved path and 
consequently possesses an angular velocity and in 
addition an angular acceleration. The axis of the 
torpedo cannot ‘‘keep up’’ with the trajectory and so 




32 


AIR FLIGHT OF A TORPEDO 


remains somewhat nose-up relative to it. As the time 
increases, the angular velocity of the trajectory de- 
creases so that the torpedo axis gets closer to it. 
There is also a damping moment acting on the axis 
of the torpedo of amount pi 6. This moment is always 
acting in the same direction since 6 is always in the 
same direction, and it tends to keep the torpedo 
nose-up relative to the trajectory. Since the trajec- 
tory is curved, d is not zero. If we were dealing with a 
straight line trajectory, 6 would be zero, and the tor- 
pedo would damp out to zero pitch. 

5.2.3 Release Conditions for Pitching Motions 

The pitch angle at release, ao, will vary from one 
airplane to another, and for a particular plane will 
depend on speed and also on the angle at which the 
torpedo is suspended in the plane. It probably will 
also depend on whether or not the airplane is moving 
horizontally at release and whether the airplane is 
accelerating, decelerating, or moving at constant 
speed. However, for roughly horizontal release and 
roughly constant speed we may say 

Weight of airplane = Lift force on plane = Kx^a plane, 

where K is a constant characteristic of the airplane. 

The attitude of the thrust line of the airplane is 
inversely proportional to the square of the release 
velocity. For a particular airplane ao varies with xd 
and with the loading. 

Figure 13, showing a few standard airplanes, is a 
graph of the angle a reference line in an airplane 
makes with the horizontal and also the angle the 
torpedo axis makes with this line when suspended in 
the plane. These curves have been obtained experi- 
mentally, but the variation of a is as expected. Con- 
sequently, for each airplane, for roughly horizontal 
release and constant speed, the value of ao is known. 

The situation with regard to ho is much more com- 
plicated. The pitching angular velocity at release 
will depend on the angular velocity of the axis of the 
torpedo and on the angular velocity of the trajectory 
Bq = g cos^ Bq/x^. However, after l)eing released, the 
torpedo falls through a region of disturbed air which 
imparts to it an additional angular velocity. 

At the instant of release the angular velocity of the 
torpedo axis depends on many factors, such as the 
angular velocity of the airplane at release, the tra- 
jectory angle, the acceleration or deceleration, the 
pitch angle at release, and also probably on the re- 


lease velocity and the weight of the airplane. The 
dependence on these factors is not known experi- 
mentally and can only be estimated theoretically. 
However, assuming the effect of most of these factors 
is known so that the pitching angular velocity at 
the instant of release is known, since the condition 
of the air beneath the plane is cpiite variable, the 
angular velocity of the torpedo when emerging from 
the small region of disturbed air is highly uncertain. 
From studying a number of photographs taken at 
the Newport Torpedo Station, showing launchings 
with various release velocities, the pitching angular 
velocity at release for a few standard airplanes was 
estimated. The average values of do are listed in the 
table below. 



Number 

do (Average pitching angular 


of drops 

velocity at release) 

TBM, TBF 

6 

-1-17 degrees per sec 

PVl 

6 

— 15 degrees per sec and -1-4 degrees 
per sec 

F7F 

7 

— 12 degrees per sec 

SB2C 

6 

-h 10 degrees per sec and — 7 degrees 
per sec 

A-20 

6 

H-IO degrees per sec and —8 degrees 
per sec 

A-26 

3 

-b 2 degrees per sec 

B-25 

4 

-j- 8 degrees i)er sec and -1-14 degrees 
per sec 

B-26 

4 

-j- 3 degrees per sec 


It should be remembered that there is a dispersion 
around the values listed in the table. Where two num- 
bers appear for do it means that a few of the observed 
launchings were found with each of the values for do. 

Thus we may say that, while ao is known for the 
airplanes now in use and may be determined experi- 
mentally for future airplanes, do is uncertain. 

From water-entry considerations it is desirable to 
have the pitch angle at entry as small as possible. 
This requires keeping ao and do small. Most airplanes 
are designed so that ao = 0 near the middle of their 
velocity range. However, since from tactical and 
other considerations it is much better to release at as 
high a velocity as possible, it seems wise, if possible, 
to have ao = 0 near the upper end of the velocity 
range for a given airplane. 

5.2.4 Release at an Angle 

In order to complete the illustration and discussion 
of theoretically predicted air trajectories, some exam- 
ples of release at angles different from zero will be 
considered. We shall consider two cases in which the 




DISCUSSION AND ILLUSTRATION OF THE THEORY 


33 


(/) 

lU 

°s 

^ UJ 


< 

u. ^ 
o 5 


UJ 

_l H 
O CO 
Z 3 

< tr 


CL 

3 

UJ 

(O 

0 

1 



F7F AIRPLANE 

TORPEDO AXIS 1*45' 
NOSE DOWN 
FROM THRUST LINE 


Ul 

o 


UJ 


O 

< H 
»- CO 
H 3 

< OC 


UJ 


UJ 
-J 

«z 
< o 


Q. 

3 

UJ 

CO 

o 

t 



TBF-3 AIRPLANE 

TORPEDO AXIS 2* 

NOSE DOWN 

FROM CENTER THRUST 
LINE 


CO 

o 

3 


^ UJ CO 

< > 

Ul 

ll cr 

O w CP 
UJ 

UJ UJ o 
_l z 


< H 


CO 

3 

q: 

X 

K 


CL 

3 

Ul 

CO 

O 

z 

t 



A- 20 C AIRPLANE 

TORPEDO AXIS 
PARALLEL TO 
THRUST LINE 


CO 

UJ 

UJ 

(T 


^ CP 
o Ul 
< o 

H 


Ll UJ 
O Z 

UJ "* 



h* 


O. 

3 


UJ 

CO 

O 


1 



140 180 220 260 300 340 380 


SB2C-5 AIRPLANE 

TORPEDO AXIS 7* 30* 

NOSE DOWN FROM 
THRUST LINE 

GROSS WEIGHT 
16,884 LBS« 


INDICATED AIRSPEED IN KNOTS 


Figure 13. Airplane attack angle in level flight versus indicated speed. 



34 


AIR FLIGHT OF A TORPEDO 


torpedo is released with a velocity of 425 knots. It is 
to be remembered that, for release, while the airplane 
has an upward component of velocity, the trajectory 
angle at release is negative, 6q < 0. In this case the 
torpedo is being “ tossed,'’ and this method of release 
is described as toss bombing. When the airplane is 
heading downward at release, Bq is positive, and the 
method of release is described as glide bombing. The 
release conditions for the two cases considered are: 

Toss bombing: 

1. To = 425 knots, Bq = —10°, height of re- 

lease = 800 ft, ao = 5°, do = 15 degrees per 
sec. 

Glide bombing : 

2. T^o = 425 knots, Bq = 10°, height of re- 

lease = 800 ft, ao = 5°, do = 15 degrees per 
sec. 

The altitudes at release were chosen to conform to 
what is thought to be the minimum height at which 
the airplane can release the torpedo for the value of 
Bo chosen. 


For all cases To (release velocity) = 425 knots ( = 


Type of trajectory 

do 

yo 

ft 

t 

sec 

Air 

-10° 

800 

12.36 

Vacuum 

-10° 

800 

11.93 

Air 

o 

O 

800 

4.36 

Vacuum 

10° 

800 

4.17 


do = trajectory angle at release. 
yo = height at release. 
t = time of drop. 
de = trajectory angle at entry. 

= entry velocity. 

R = horizontal range in yards. 

If the airplane is traveling in an upward circle at 
release, the pitch angle will be more nose-up than 
for horizontal release, and vice versa for a circle 
concave-downward. 

In Figure 14 the trajectory for case 1, the vacu- 
um trajectory for case 1, and the pitching oscilla- 
tions are shown. (The trajectoiy without the Mark 
1 drag ring would be intermediate between the 
vacuum trajectory and the trajectory with the drag 
ring.) 

Fh’om Figure 14 it is readily seen that the effect of 
the drag forces is similar to horizontal release and is 
even more marked because essentially they are acting 


for a longer time than a torpedo having the same 
entry conditions but released horizontally. 

Thus the range is less than that of the vacuum tra- 
jectory. In addition, due to the long time of flight, 
the pitching oscillations have markedly damped out 
so that the pitch angle at entry relative to air is 
very small. 

From Figure 15 it is also seen that the range is less 
with the drag forces acting than with a vacuum tra- 
jectory. For this trajectory, even though the time of 
flight is small, the pitch oscillations have had time to 
damp out almost completely at entry. Due to the 
high release velocity the damping of the pitch oscil- 
lations is large in both trajectories and at entry there 
appears the small nose-up pitch. 

A comparison of the two trajectories and the 
vacuum trajectories may be made from the table of 
release conditions, entry conditions, height at release, 
and time of flight. Also in the table are the conditions 
for a horizontal release which would have the same 
Be and Ve with the range and time of drop for such an 
“equivalent horizontal release.” 

From this table the advantages of a toss bombing 


08 ft : 

per sec), ao 

p' 0 • 

= — o , ao 

— 1 c; 

degrees per 

sec. 


de 

TV 

ft per sec 

R 

yds 

y 0 

y'o 

t' 

R' 

26.4 

486 

2,488 

633 

1,026 

8.70 

1,671 

20.1 

754 

2,812 

707 

1,042 

8.05 

1,898 

21.2 

620 

983 

816 

998 

8.53 

1,034 

20.1 

754 

1,025 

707 

1,026 

8.05 

1,939 


T'o = velocity when trajectory is horizontal (release velocity 
for equivalent horizontal release). 
y'o = height when trajectory is horizontal (height for equiva- 
lent horizontal release). 

V = time of drop for equivalent horizontal release. 

R' = horizontal range for equivalent horizontal release. 

method of release is clear. Thus with the Mark 13 
torpedo the enormous range in air of 2,488 yd can 
be achieved with an entry angle of only 26.4°. The 
time of flight is not prohibitively large and the pitch 
angle at entry relative to air is very small. This is 
important from the point of view of water entry. It is 
clear that the vacuum trajectory will have a greater 
range (yards), a shorter time of flight, and a smaller 
entrance angle. The Mark 13 without a drag ring 
will have intermediate values. The very large range 
obtained by a toss bombing method of release (with 
very similar entry conditions to those for horizontal 
release) is of significance in tactical considerations. 


DISCUSSION AND ILLUSTRATION OF THE THEORY 


35 



HORIZONTAL RANGE IN FEET 



Figure 14. Trajectories and pitch oscillations for toss bombing release, torpedo tossed at Oq = —10°, Fo = 425 knots, 
released with ao = —4°, do = 15 degrees per sec. 



Figure 15. Trajectories and pitch oscillations for glide bombing release, torpedo gliding initially at 0o = —10°, 
Fo = 425 knots, released with ac = — d°, q:o= 15 degrees per sec. 


36 


AIR FLIGHT OF A TORPEDO 



TIMES IN SECONDS 

Figure 16. Yaw angle versus time from release \pQ = 3°, = 0 degrees per sec. 



TIME IN SECONDS 

JdGURE 17. Yaw angle versus time from release xpa = 0°, ypo = 15 degrees per sec. 


For the glide bombing method of release, the hori- 
zontal range is comparatively small and pitch angle 
at entry may still be comparative!}^ large. Obviously 
the only possible advantage for this method of release 
is the tactical one of a short time of flight. 

5.2.5 Yawing Motion 

In order to discuss the yaw oscillations, Figures IG 
and 17 have been plotted. In Figure IG, using the 
constants for the Mark 13 torpedo, Mark 2-1 stabi- 
lizer and Mark 1 drag ring, two curves of the yawing 
motions are drawn. In Figure IG, = 3° and po = 0 
with one curve drawn for Xo = 200 knots and one for 


Xq = 350 knots, while in Figure 1 7 corresponding to 
these two release velocities curves are plotted for the 
release conditions po ^ 0 and po = 15 degrees per sec. 

These curves are seen to be ordinary damped har- 
monic oscillations about a mean yaw angle of zero 
degrees. From these curves we notice, as in the pitch 
oscillations, the amplitude of the yawing motion is 
more rapidly damped out as the release velocity is 
inci'eased. Furthermore, it should be noted that the 
yawing motion is independent of the release angle 6o, 
except insofar as 6o changes the time of drop. 

From these curves the effect of an initial yaw angle 
and yawing angular velocity also can be seen. For 
present torpedo launchings the pilot must head the 


DISCUSSION AND ILLUSTRATION OF THE THEORY 


37 


airplane in the direction he wants the torpedo to run 
and attempt to release the torpedo so that t/'o = 0 and 
\po = 0. These conditions must be observed because 
the gyro is unlocked approximately 0.5 second after 
the torpedo is released. From Figure 16 it is seen 
that, if the plane is not heading in the right direction 
but is at yaw angle xpo = 3 degrees, then the gyro 
can be off b}" 1.4 degrees when it is unlocked, while, 
if the airplane is spinning around to a lead angle and 
releases when xpo = 15 degrees per second, it is seen 
from Figure 17 that the gyro can be off by 5.3 degrees. 
Thus it is clear why the pilot must attempt to main- 
tain \po = = 0. Furthermore, it is clear that there 

would be advantages derived from releasing the gyro 
just before the torpedo is released. 

5.2.6 The Effect of Roll 

We have made no mention thus far of the rolling 
angular velocit}" of the torpedo during the air flight. 
The roll angle is the angle through which the torpedo 
rotates about its longitudinal axis. This angle is 
indicated by 0 and the rolling angular velocity by 0. 
A torpedo may have a rolling velocity in air due to 
the release conditions. Thus the surface of the slings 
which release the torpedo is not smooth and, since 
the slings are released on one side first, due to fric- 
tion the torpedo rolls off. Also, the slipstream beneath 
the airplane which acts on the stabilizer might in- 
duce a roll velocity. The effect of the slings may be 
minimized by greasing them or by the use of a single 
suspension bar. At the Newport Torpedo Station it 
has been observed that this bar releases the torpedo 
with very little roll velocity. 

During the drop the roll angular velocity of the 
torpedo is practically constant since there is almost 
no damping of the motion. Due to the moments 
acting on the torpedo and the pitching and yawing 
motions, if the torpedo has a roll velocity, it must 
exhibit a gju'oscopic effect. The magnitude of this 
effect has been calculated. The gyroscopic effect on 
roll or yaw should rarely be more than 0.17 degree 
at any time during the drop and for most launchings 
should not be more than one-quarter of this value. 

The primary effect of the roll in air is due to the 
asymmetry of the stabilizer. Thus co is different from 
0 )' and the resulting motion is altered due to the 
change in the constants in any particular plane. This 
will be discussed further in a later section. 


5.2.7 Roll Stabilization 

Some torpedo air stabilizers, notably some em- 
ployed by the British, Germans, Italians, and Jap- 
anese, possess gyro-controlled ailerons which serve 
to prevent large roll in air. It is necessaiy to prevent 
roll in air by such a device if it is desired to gain some 
particular advantage of an asymmetrical torpedo 
head at entrv or if it is desired to have a torpedo 
enter the water with some mean pitch angle other 
than zero. For example, if the air stabilizer is preset 
so that the mean pitch angle is some amount nose-up, 
the torpedo must be prevented from rolling over in 
air since the nose-up pitch angle will become a yaw 
angle when rolled 90 degrees and will become a nose- 
down pitch angle if the torpedo rolls 180 degrees. 

The gyro-controlled ailerons produce a torque 
about the longitudinal axis of the torpedo which is 
proportional to the square of the velocity (F“) and 
for moderately small roll angles to the roll angle itself 
0; thus the torque is qo4>, where qo is proportional 
toF“. In addition, there is a damping moment which 
is probably small and is proportional to the velocity 
and to the rolling angular velocity 0; the damping 
moment is then po0, where po is proportional to V. 

Thus the equation of motion is 

C0 -f- Po0 + ^00 = 0 , (26) 

where C is the moment of inertia of the torpedo about 
a longitudinal axis through the center of gravity. For 
the Alark 13 torpedo, C = 30.4 slug ft^. 

The roll angle as a function of time is then simply 

0(0 = ^00 cos wt 4- ~f _ p0o 2 

where 


po q 

P = — , CO = \ . 

C 4(72 

Generally, for most torpedo releases, 0o = 0, and 
00 5^ 0. The values of qo/V~ and po/V depend both on 
the size and position of the ailerons as well as the 
torpedo. po/V is probably small. 

This motion is seen to be that of ordinary damped 
oscillations. If 0o = 0, as appears to be the usual con- 
dition, the roll angle is always less than 0o/oo. 


38 


AIR FLIGHT OF A TORPEDO 


5.2.8 Other Approximations 

For all the illustrations the complete expressions 
were used. For many applications, however, this is 
not necessary. 

For quantities associated with the trajectory, the 
series expressions are usuall}^ sufficiently accurate 
and may be easier to use. For many purposes the 
term may be neglected. The succeeding terms be- 
come less significant the smaller the value of kxot. For 
all practical applications the expressions derived are 
sufficiently exact. The number of terms of the ex- 
pressions to be retained in any problem depends on 
the particular application. 

The complete expression (22) has been used for the 
illustrations of the pitching motion. However, for 
many applications the particular solution may be 
neglected,^ and the solution of the homogeneous 
equation given in equation (20) should prove suffi- 
ciently accurate. From the illustrations given it ap- 
pears that the particular solution is rarely more than 

0.8 degree nose-up, the mean value rarely more than 
0.4 degree nose-up, and they both decrease with time 
and also decrease rapidly with increasing release 
velocity. Consequently, using only the solution of the 
homogeneous equation is more accurate the greater 
the release velocity. However, again, the accuracy 
required of the solution depends on the particular 
application. 

5.3 DISCUSSION OF EXPERIMENTAL 
RESULTS AND COMPARISON 
WITH THEORY 

5.3.1 Comparison of Pitch Oscillation 
Theory and Experiment 

Experimental curves of pitch angle against alti- 
tude or time can be obtained from the photographs 
of torpedo launchings at the Newport Torpedo Sta- 
tion. Some of these experimental curves have been 
matched by inserting a])propriate values of p and co 
into a slightly modified form of ecpiation (22). From 
the experimental curves, ao and do were obtained at 
a particular time, and, since p and co were unknown, 
different values were tried until a good matching of 
the experimental curve l)y theory was obtained. 
Figures 18, 19, 20, and 21 are graphs of experimental 
curves obtained at the Newport Torpedo Station and 

^ This has V)een done in the appi’oxiniate treatments given in 
reports issued by the U. 8. Navy Torpedo Station at Newport 
and the British. i 


of the curves fitted to them on the basis of the theory 
given in Section 5.1. 

The experimental curves are based on photographs 
of airplane launchings. Since in most airplanes, espe- 
cially those which have an internal installation of the 
torpedo, it is not possible to see the torpedo clearly 
until it has fallen for 0.5 sec (about 4 ft), the points 
on the experimental curves of pitch angle are really 
not known until about 0.5 second after release. Con- 
sequently, in ecpiation (22) the initial conditions of 
the motion, which are specified l^y ao and do, were 
taken about 0.5 sec after release. Since the trajectory 
angle at release is zero and t = 0 was assumed to be 
about 0.5 sec after release. Bo, the trajectory angle at 
the assumed / = 0 is given by 

^0 ~ tan do = 0.5Ci = . 

i'o 

For such a short time the single term gives sufficient 
accuracy. 

In the graphs there is a close agreement between 
theory and experiment. There seems to be an indica- 
tion of the fact that the torpedo tends to damp out 
to a slightly nose-up pitch angle for all usual times of 
drop. It is unfortunate that, due to the nature of the 
camera used, photographs of launchings from alti- 
tudes greater than about 450 ft could not be made. 

To indicate whv it was necessarv to use a slightlv 
modified form of equation (22), as well as to explain 
the lack of complete experimental verification of the 
theory of the air trajectory of the torpedo, the errors 
in the experimental data must be considered. 

1. From an examination of a number of cases, it is 
estimated that there is a probable error of about 0.03 
sec in the time of drop. 

2. The probable error in determining the height of 
drop, which is determined from the time of drop, is 
about 4 ft. 

3. The errors in determining the horizontal and 
vertical velocities relative to air during the drop are 
a result of the error in determining these velocities 
relative to ground (as obtained by the camera, in 
analyzing the photographs) together with the error 
in the wind velocity value which is used to convert 
the velocities relative to ground to those relative to 
air. 

a. There is a probable error of about 3 knots in 
determining the horizontal and vertical 
locities relative to ground at any time during 
the drop. This error is due primarily to the 


COMPARISON OF RESULTS AND THEORY 


39 


probable error of about 2 per cent in determin- 
ing the scale factor of the photographs. This 
estimate of the error was obtained by meas- 
urements of the small image of the torpedo, 
b. The wind velocity for each launching at the 
Newport Torpedo Station is obtained by an 
anemometer on Gould Island at an altitude 
of 140 ft. A study of the distribution of wind 


4 


r 

oT 

UJ 
u 
cc 
o 

UJ 

o 
z 

UJ 
-I 

o 
z 
< 

z 

c 

t -4 


-6 


.2 


•6 
















-e— OBSERVED POINTS 

AND CURVE 

THEORETICAL CURVE 

NTS No. A -32 

•i . . 007 KNflTt; 





















•X 






















AIRPLANE SB2C-I 































Y 







/ 


A' 











• 

A 







J 





















ft 

1 




\ 













V 





t 





















// 






'' 

V , 














/ 




















/ 













~T 






















t- 

































































0.4 0.8 1.2 1.6 2.0 2.4 

TIME IN SECONDS 


2.8 


3.2 


3.6 


4.0 


4.4 


Figure 18. Theoretical and observed graph of pitch 
angle versus time 



1 



n 


—©—OBSERVED POINTS 

AND CURVE 
THEORETICAL CURVE 

NTS No. A-160 

X«I90 KNOTS 

AIRPLANE TBM-IC 
















































5n> 

k 











’ 


i 









N 









~T 



\\ 






/ 












/ 










TJ 












/ 









If 











1 


/ 





\j 





jJ 













r 








A 










U 

, 


/ 





— \o 

\ 

















/ 




























































t 







1 


Y 4 

(/) 

UJ 

UJ 

q: 2 
o 


o 

z 


-2 


-4 


0 0.4 0.8 


1.2 


6 2.0 2.4 2.8 

TIME IN SECONDS 


3.2 


3.6 


4.0 


4.4 


4.8 


Figure 19. Theoretical and observed graph of pitch 
angle versus time. 


velocity with height at Quonset Naval Air 
Station, which is five miles from Gould Island, 
during the months from September 1943 to 
September 1944, showed that the wind magni- 
tude increases rapidly with altitude in the first 
700 ft above the ground and that it veers in a 
clockwise direction. The effect is especially 
marked in the early morning hours, 0700, and 
diminishes towards noon as the wind strati- 
fication is destroyed by convective mixing. 

It should be noted that the balloon data 
give the average wind through the first 700 ft 


above ground and that, therefore, the maxi- 
mum wind in each layer may be considerably 
in excess of the average value. 

Due to this variation in the speed and di- 
rection of the wind with altitude, the error in 
horizontal velocity due to the error in the wind 
data may be about 3 knots. 
































































































































j/ 






















if 

f 




j\ 






/ 











/ 





Y 













« 










A 




/ 






















7 




















/ 

1 





















/ 












—©—OBSERVED POINTS 

AND CURVE 

THEORETICAL CURVE 

NTS No. A-44 

Xo = 200 KNOTS 

AIRPLANE SB2C-I 



/ 









































q: 

o 

UJ 

o 


i-2 

< 

X 

o 

t -4 


-6 


-8 


0.4 0.8 


1.2 


1.6 2.0 2.4 2.8 3.2 3.6 4.0 

TIME IN SECONDS 


4.4 


Figure 20. Theoretical and observed graph of pitch 
angle versus time. 



Figure 21. Theoretical and observed graph of pitch 
angle versus time. 


4. The probable error in determining the trajectory 
angle 6, at any time during the drop, appears to be 
about 0.6 degree. This error is due to the error in the 
vertical and horizontal velocities relative to air. 

5. The errors in obtaining the pitch angle relative 
to air at any time during the drop lie in the deter- 
mination of the trajectory angle 6, and the measure- 
ment of the angle of inclination of the axis of the 
torpedo at that time. 


40 


AIR FLIGHT OF A TORPEDO 


Since the probable error in 6 was found to be about 
0.6 degree and the error in the measurement of the 
angle of the axis of the torpedo was found to be about 
0.3 degree, the probable error in the pitch angle rela- 
tive to air at any time during the drop is approxi- 
mately 0.7 degree. 

While the probable error in the pitch angle relative 
to air at any time during the drop is about 0.7 degree, 
the error in the smooth curve (in the statistical sense) 
through the points is probably very much less. 
Nevertheless, although the probable error in the pitch 
angle may be less than 0.7 degree, the actual error 
ma}^ be very much larger. 

« 

5.3.2 Discussion of Experimental Verification 
of Theory of Pitching Motion 

In the analysis of the photographs a mean hori- 
zontal deceleration and vertical acceleration are ob- 
tained. From Section 5.1 it is seen that the trajectory 
cannot be described exactly by this method. Since 
constant accelerations are used, the values of tan 6 
obtained do not correspond exactly to the expressions 
given in Section 5.1. In matching the experimental 
curves, the expressions for 6 obtained from analysis 
of the photographs were used in each case. In this 
sense a slightly modified form of the theory was used. 
In general, there appears to be a good verification of 
the theory of the pitch oscillations given in Section 
5.1. 

However, examining many of the experimental 
curves of pitch oscillations reveals that some of them 
do not have the form predicted by the theory. Thus, 
if there were an error in the horizontal velocity, the 
trajectoiy angle 6 would always be in error during 
the drop, and the calculated line of zero pitch on the 
curves would really not be zero pitch as assumed in 
the plotting. For example, if there were an error in 
the wind measurement and the actual tail wind were 
10 knots greater than what was recorded, the pitch 
oscillation curve when plotted by the method out- 
lined would appeal' asymmetrical. The torpedo would 
appear to favor a nose-down pitch since the ampli- 
tude of the oscillation in the nose-down direction 
would be larger in magnitude than the preceding 
amplitude in the nose-up direction, which is impos- 
sible according to the theory given in Section 5.1. 
There are (piite a few experimental curves which 
appear to exhibit this type of error. It should be re- 
membered that this is an error in the experimental 
pitch angle curves and the torpedo does not oscillate 


with respect to the air in the manner indicated by 
the curves. Since these curves do not represent the 
actual oscillations of the torpedo, they cannot be 
matched by theory. 

Similarly, an error may often arise (which has 
somewhat the same appearance) due to the fact that 
the Mark 2-1 stabilizer or one flap of the stabilizer 
may have rocked during the air flight. This type of 
motion will represent the actual oscillations of the 
torpedo since the stabilizer behaves like one which is 
preset. By fixing the stabilizers this motion has been 
produced at the Newport Torpedo Station. These 
curves can be matched by the theory in Section 5.1 
simply by adding to the equation (22) a constant 
mean pitch angle, which must be known, however. 

Some irregularities in the curve may be due to the 
effect of the variation of the wind magnitude on co 
and p. Thus, if the wind changes a certain amount, 
o) and p, which are proportional to the velocity, will 
vary accordingly, and the actual oscillations will not 
be periodic or described exactly by a theory which 
assumes a constant co and p. However, the magni- 
tude of this error is generally very small. 

Again, as has been noted in Section 5.4.2, the co in 
the pitching motion is affected by roll. This will alter 
the experimental curves. The quantitative effects 
will be discussed further in Section 5.4. 

5.3.3 Discussion of Experimental Verification 
of the Theory of the Trajectory 

As has been mentioned, the method used in the 
analysis leads to a mean horizontal deceleration and 
vertical acceleration. These cannot be the actual de- 
celeration and acceleration. Nevertheless, for the 
short times of drop that are involved in the photo- 
graphed launchings and for the relatively low release 
velocities, the terms involving higher powers of t 
than are negligible. The average time of drop was 
only some 4.6 seconds and the average speed some 
175 knots. Neglecting the terms, the horizontal 
acceleration is —kx^^ + 2kHoH, and the vertical ac- 
celeration is ig — kiVdxo) — gkxot. In a time T, the 
average values of these would be —kxo^ + kH^^T 
and {g — kiVaXi^) — (gkxo/2)T. The maximum errors 
in the accelerations produced by using these mean 
expressions are, at time T, kH^^T and — (gkxo/2) T. 

For the photographed drops the average value of 
T = 4.5 sec and Xo = 296 ft per sec. Consequently, 
the order of magnitude of the maximum error in the 
horizontal deceleration of 0.23 ft per sec per sec or 


COMPARISON OF RESULTS AND THEORY 


41 


about 5 per cent of the horizontal deceleration ob- 
tained by using only the f term, and in the vertical 
acceleration the maximum error is 0.95 ft per sec 
per sec or about 3 per cent of the acceleration given 
by the term. These are small errors because the 
times of drop and release velocities are relatively 
low. However, for greater speeds and altitudes of 
release the error involved in using the average accel- 
eration is not negligible. 

In view of the previous discussion, since t and Xq 
are both relatively small for photographed launch- 
ings, the higher powers of t than f in the expressions 


using for k the slope of the straight line in Figure 22 
plus 5 per cent of the slope. 

Again, in the y or vertically downward direction, 
because of the relatively small times of drop and 
release velocities, the effect of terms cannot be 
noticed. From the analysis of the photographs the 
mean vertical acceleration is obtained. We expect the 
slope of the graph of y{t) obtained from the photo- 
graph to decrease somewhat with time, but, since t 
and Xt) are small, the effect is only a change in slope 
of about 1.90 ft per sec per sec or about 6 per cent 
change in slope, which is hardly observable. 



Figure 22. Horizontal deceleration versus (horizontal release velocity)^ for the Mark 13-2 torpedo fitted with Mark 
2-1 stabilizer and Mark 1 drag ring. Correlation coefficient = 0.94. 


for the trajectory in Section 5.1 are small and conse- 
quently difficult to observe. Also they are usually 
masked by the experimental errors involved in 
analyzing the photographs. Consequently, it is ex- 
tremely difficult to attempt to verify the higher 
power terms than in the expressions for the air 
trajectory. In spite of this, the method used to ob- 
tain k serves as some source of experimental verifica- 
tion. In order to determine k, a graph was plotted of 
the mean horizontal deceleration against Xo^. Since 
the effect of the term is small for this application, 
we should expect a good correlation between the 
deceleration and Xo^ from the plot, and the slope of 
this graph should give the value of k. Figure 22 shows 
this. There is a correlation coefficient of 0.94, which 
indicates the correctness of the theory. The effect of 
the term is approximately taken into account by 


One might expect that, using the average value of 
the time of drop and release velocity, we might hope 
to obtain the average vertical acceleration and com- 
pare the result with the experimental averages of the 
launchings. This method breaks down due to the fact 
that, while the pitch oscillations about the mean 
value in any duration of time tend to cancel out to 
zero, actually in a relatively short time of drop there 
is not a complete cancellation. Thus the theory of 
Section 5.1 predicts an average vertical acceleration 
for the observed launchings of about 3 1. 1 ft per sec 
per sec, while the average results yield the value of 
31.5 ft per sec per sec. This slight difference may be 
explained by the fact that, for the airplanes used, 
the average pitch angle at release is about 3 degrees 
nose-down so that the net result in the relatively 
short time of drop will tend to be some small average 


42 


AIR FLIGHT OF A TORPEDO 


pitch angle nose-down which will produce a lift force 
downwards and, consequently, could increase the 
average vertical acceleration from 31.1 ft per sec per 
sec to 31.5 ft per sec per sec. 

5.4 AERODYNAMIC CONSTANTS OF 
THE MARK 13 TORPEDO 

Thus far we have developed the theory of the air 
flight of the torpedo and discussed some of the experi- 
mental evidence that tends to support the point of 
view presented. We now proceed to the determina- 
tion of the constants associated with the Mark 13 
torpedo with its various air appendages. 

5.4.1 Method of Obtaining Aerodynamic 
Constants and Tabulation 
of Results 

The characteristics of the Mark 2-1 and Mark 2 
stabilizers, with and without the Mark 1 drag ring, 
have been olitained by wind tunnel tests at various 
places including the Newport Torpedo Station, 
Wright Field, Langley Field, and the University of 
Michigan. In addition, strip camera photographs of 
airplane launchings at the Newport Torpedo Station 
can be interpreted in terms of the theory to give the 
same constants. 

The model tests at the Newport Torpedo Station 
were carried out on a 3^- size model at a velocity of 
100 ft per sec. At Wright Field a 3^-size model was 
used at a velocity of 100 miles per hour. A full-size 
model was used at Langley Field. The Llniversity of 
Michigan tests were carried out on a 0.2236-scale 
model, with a wind velocity of 88 ft per sec. All these 
models were tested without propellers. 

From the wind tunnel tests it is possible to deter- 
mine Cm/\P, C l, a., and Cd- However, 

with the facilities available, Ck and Cp could not be 
obtained in the tunnels. One result for Ck was ob- 
tained with an oscillating model, but otherwise these 
constants have not been determined from tunnel 
tests. 

The method of obtaining Cm ' a, Ck, and Cd from 
strip camera photogi'aphs may be described as match- 
ing. Using experimentally determined curves of a{t) 
and using the value of a and the slope d at a par- 
ticular time as the initial conditions of the motion, 
different values of p and co were used in the modified 
form of equation (22) until a good matching of the 


experimental curve by the theoretical curve was ob- 
tained. Thus, from curves like those illustrated in 
Figures 18, 19, 20, and 21, where experiment is 
matched by theory, the values of p/V and^ V~ were 
tabulated. 

From p/V and q/V- and equations (4) and (5), Ck 
and Cm / a are obtained. 

In order to obtain Cd, the horizontal decelerations 
as recorded at the Newport Torpedo vStation (ob- 
tained from the strip camera photographs) were 
plotted against Xo^ and the best straight line fitted 
with the condition that it pass through the origin. 
This is plotted in Figure 22. If the f term is negligible, 
the slope of this line is the value of k. The average 
contribution of the term to the deceleration in the 
cases considered is about 5 per cent of the C term. 
Consequently, to the slope obtained from Figure 22, 
5 per cent was added in order to find the value of k 
and hence Cd- 

The following constants were used to describe the 
Mark 13 torpedo fitted with a shroud ring, Mark 2-1 
stabilizer, and a drag ring. 

/ = moment of inertia about a transverse axis 

through the center of gravity = 972 slug ft-. 

p = density of air = 0.002378 slug per fU. 

11^ = weight of the torpedo = 2,200 lb. 

I = length of the torpedo = 13.4 ft. 

We may then write CM/ct, Ck, Cd, etc., in terms of 
q'V^, p/V, k, etc., and these constants. 

The results obtained by the various methods are 
listed in Table 1. 

5.4.2 Discussion of Aerodynamic Constants 

Before proceeding to discuss the constants ob- 
tained by the various methods, the effect of propellers 
on the coefficients must be pointed out since this may 
account for a good part of the difference between 
tunnel tests and full-scale results obtained by match- 
ing. All the model tests were run without propellers. 
It has been found in wind tunnel tests that the re- 
storing moments are markedly increased by the addi- 
tion of propellers. In recent tests at the Newport 
Torpedo Station similar results were obtained over a 
large range of pitch angles. It is also found that the 
addition of propellers increases the damping coeffi- 
cient considerably. Hence, due to the effect of pro- 
pellers, it is to be expected that results from full- 
scale matchings should be somewhat different from 
results obtained in wind tunnels. 


AERODYNAMIC CONSTANTS OF THE MARK 13 TORPEDO 


43 


Restoring ^Moment Coefficient Cm/ a 

From the values of Cm/cc presented in Table 1, it 
appears that the values obtained by matching ex- 
perimental curves, which represent the full-scale 
torpedo in flight, are usually more negative than the 
values obtained in the wind tunnels. This is explained 
to some extent by the absence of propellers in the 
model tests. Furthermore, it is noticed that the 
]\Iark 1 drag ring seems to increase the negative 
value of C.u/q:. 


so that one might expect a dispersion in the values of 
Cm/cc^ The value of Cm/ a. = 1.50 is, therefore, prob- 
ably the value at some average roll angle other than 
0 degrees. 

For the Alark 2 stal^ilizer the area of the smaller 
face is much less than in the Mark 2-1 stabilizer so 
that the effect of roll in air on Cm/oc will be greater 
than in the Mark 2-1. Consequently, the value of 
Cm/ a will be too low, due to roll, by a larger factor 
than is expected from a comparison of the areas of 
the stabilizers in the pitching plane. That this does 


Table 1. Results obtained for the constants of the Mark 13 torpedo with shroud ring fitted with Mark 2-1 and Mark 2 
stabilizers with and without the Mark 1 drag ring (pickle barrel) by matching experimental curves with theoretical 
curves and by the wind tunnel tests at the Newport Torpedo Station (N), the University of Michigan (M), Wright 
Field (W), and Langley Field (L). All angles are in radians. 



Mark 2-1 with Mark 1 drag ring 

Mark 2-1, no drag ring 

Mark 2 with drag ring 


Matching 

Wind tunnel 

Matching 

Wind tunnel 

Matching 

Wind tunnel 

CmI^ 

-l.oOdzO. 13 

-1.19(X) 
-1.39 (W) 
-1.3G(M) 

— 1 . 25 ±0. 17 

-1.16 (L) 

-1.16 ±0.03 

-1.16(X) 

C K 

-2.87db0.0o 


1 .31 ±0.39 


2.43±0.11 

2.32 (P) 

0.87 (X) 
0.97 (W) 
0.84 (M) 

1 . 70 ±0 . 55 

0.32 (X) 
0.38 (W) 
0.30 (M) 

C D 

0.93 






Cjcc 

C.\f / 


4.8 (X) 

-0.717 (X) 
0.774 fM) 

4.9 (L) 

-0.662 (X) 










The value of C m/ol obtained by matching is sub- 
ject to a probable error of about 10 per cent. The 
main cause of this dispersion seems to be the roll of 
the torpedo in air. During some part of the drop, the 
small face of the Mark 2-1 or Mark 2 stabilizer may 
have rolled into the pitching (vertical) plane, and 
the effective value of Cm/ a would therefore he dim- 
inished. The Mark 2-1 stabilizer is not symmetrical, 
and the value of Cm/ a varies with the angle of roll. 
From wind tunnel tests at the Newport Torpedo 
Station it was found that, for a stabilizer very similar 
to the Mark 2-1, when the torpedo rolled 45 degrees 
the value of Cm 'ol was 0.82 times the value at 0 
degrees roll and, when the torpedo rolled 90 degrees 
it was 0.05 times the value at 0 degrees. It was not 
possible from the photographs to ascertain the magni- 
tude of the roll of the torpedo during the air flight 


not appear as an increased dispersion ma}^ be due to 
the small number of observations. 

Damping Moment Coefficient Ck 

As has been pointed out, elaborate arrangements 
are recpiired in order to measure Ck in wind tunnels; 
consecpiently, very few results have been obtained. 
The drag ring probably produces damping of the 
oscillations of the torpedo by dissipating energy in 
shedding vortices. The damping effect of the sta- 
bilizer is due to the additional pitch angle at the tail 
of the torpedo caused by rotation. The moment due 
to the stabilizer caused by the rotation will therefore 
be affected by roll in the same way as Cm/ ol is altered 
by tlie roll in air. Therefore, roll will contribute to 
causing a dispersion in CV, and this should be more 
marked without the drag ring since the drag ring 




44 


AIR FLIGHT OF A TORPEDO 


contributes a constant amount to Ck- In addition, 
the mean value of Ck for the Mark 2 stabilizer may 
be expected to be lower than for the Mark 2-1 since 
the effect of roll is larger because of the greater differ- 
ence of the areas in the pitching and yawing planes. 

In Table 1, two values of Ck are given for the Mark 
2-1 stabilizer without a drag ring. This is due to the 
fact that very large values of Ck obtained by match- 
ing raised the average value above what is probably 
the true value. It was found from an inspection of a 
number of curves of a(t) for launchings without drag 
rings that Ck is small so that the average value of Ck 
without a drag ring is in the neighborhood of 1.31. 
The dispersion in the value of Ck without a drag ring 
is considerably greater than with the drag ring, even 
for the value Ck = 1.31. This is to be expected from 
the discussion above and is explained by the roll in 
air. It is noticed that Ck with a drag ring is more than 
twice the value without it. This effect will be illus- 
trated later. 

Drag Coefficient Cd 

The value in Table 1 was obtained from the graph 
in Figure 22 as has been pointed out earlier. This 
value is probably the most reliable one since it is 
based on full-scale actual flight tests of the torpedo. 
It is in fair agreement with wind tunnel results. The 
effect of propellers on Cd is not expected to be im- 
portant, and this expectation appears to be con- 
firmed. 

Lift Coefficient C l/ cl 

This coefficient could not be obtained from analyz- 
ing strip camera records or photographs of airplane 
launchings. The photographs could not give a pre- 
cise value for Cl because the times of drop were 
relatively low and the lift effect on the Mark 2-1 
stabilizer is very small and masked by the pitch 
oscillations. Consecpiently, the only results for C l! cl 
are wind tunnel measurements. 

Damping Force Coefficient C f 

Without the Mark 1 drag ring, Ck = 1.31. With- 
out the drag ring, most of the damping arises at the 
tail of the torpedo. This also has been found true for 
airships. Consequently, the damping force is con- 
centrated at the tail, and the damping force coeffi- 
cient is Cf = 13.4/().7 = 2(7/v, where (>.7 ft is ap- 
proximately the distance from the center of pressiu’e 
of the air forces acting on the torpedo to the center 
of gravity. Now, when the torpedo is fitted with a 


drag ring, we find Ck is about doubled. Since most of 
the forces due to the drag ring, which produces this 
additional damping moment, are probably concen- 
trated at the nose, when Ck is doubled, the damping 
forces at the nose must be roughly equal to those at 
the tail and in the opposite direction. Consequently, 
a couple is produced so that with the Mark 1 drag 
ring it seems probable that Cf = 0. 

Moment Coefficients in the Horizontal Plane 

The values of Cm/\P and Ck in the horizontal plane 
cannot be obtained by matching experimental curves 
since, as yet, there are no accurate observations of 
yaw angle as a function of time in the horizontal 
plane for the Mark 13 torpedo with air appendages. 

For Cm/\1/ we can use the wind tunnel results to 
the extent of learning that Cm/\P is about 0.6 times 
Cm/ a so that taking 0.6, the value of Cm/ a as ob- 
tained by matching experimental curves, we have 
Cm '^ = 0.9. Ck in the horizontal plane is probably 
close to the value in the vertical plane. Although the 
area of the stabilizer effective in the horizontal plane 
is less than in the vertical plane, the effect of the edges 
in the horizontal plane should produce some damping 
which may result in about the same Ck as in the 
vertical plane. In addition, the part of Ck due to the 
drag ring is present in the same amount in the hori- 
zontal as in the vertical plane. Again, Cf should be 
about the same as in the vertical plane since Ck is 
about the same, that is, Cf = 0 in the horizontal 
plane, while without the drag ring Cf = 2Ck- For 
ClCA there are no experimental results. However, 
since Cm/^' = 0.6C,u/ «, one may estimate that 
Cl/V = 0.6Cl/ a. 

5.4.3 Characteristic Aerodynamic Constants 
of the Mark 13 Torpedo with 
a Shroud Ring 

Using the values obtained from actual air flight 
tests wherever possible, the following appears to be 
a reasonable set of constants. 

1. Fitted with IMark 2-1 stabilizer and Mark 1 drag 
ring (pickle barrel) 

C,,/a = -1.50, 

Ck = 2.87 (pitching and yawing), 

Cd = 0.93, 

Cf =0 (pitching and yawing), 

(h/a = 4.8, 

Cm/^l = 0.90, 

Cl/^P = 2 . 88 , 


QUANTITIES RELATIVE TO AIR AND TO GROUND 


45 


1 

.4 

M 

I 

p 


== 972 slug ft-, 

= 2.75 

2200 , 

= -slugs, 

32.16 

= 13.4 ft, 

= 0.00238 slug per foot^. 


The constants in the equations of motion become 

CO = 13.8 X 10“^ V (knots) sec~\ 

p = 2.94 X 10“^ T" (knots) sec~\ 

k = 4.47 X lO-^ft-i, 
r = 14.11 X 10~^T" (knots) sec“\ 


where T' (knots), may be obtained from Figure 7. 
These are the constants used in constructing the 
nomogram in Section 5.2.1, Figure 9. 

2. Fitted with Mark 2-1 stabilizer, no drag ring, 

C^f/a = 1.25, 

Ck =1.31 (pitching and yawing), 

Cd = 0.33, 

Cf = 2.62 (pitching and yawing). 

Cl/ O' = 4.9, 

CM/^p = 0.75, 

Cl/xP = 2.94, 

CO = 12.6 X 10~n" (knots) sec“\ 

p = 1.34 X 10~^F (knots) sec“\ 

k = 1.60 X 10-Mt-i, 
n = 12.7 X 10~n^ (knots) sec“h 

It is both important and interesting to note the 
effect of the Mark 1 drag ring on the air flight of the 
torpedo. Besides the increased drag in air which may 
cause the velocity of the torpedo at entry to be as 
much as 40 knots less than the release velocity, the 
importance of the drag ring is due primarily to a large 
increase in the damping of the pitch oscillations. 
Consequently, for the same release conditions, tor- 
pedoes fitted with drag rings, in general, will have a 
much smaller pitch angle at entry than torpedoes 
without drag rings. The amplitude of the pitch angle 
at any time t without the drag ring divided by the 
amplitude with the drag ring is given roughly by a 
factor of the type 


g - 1.34 X 10-3 Vt 
g - 2.94 X 10-3 Vt 

which may be seen to be of considerable importance 
in determining the subseciuent underwater tra- 
jectory, as is shown in Chapter 6 (see end of Section 
6.3). Incidentally, this ratio increases with time. 


This shows the great effect of the Mark 1 drag ring 
in changing the pitch oscillations and hence improv- 
ing the water entry. 

It should be noted that the shroud ring does not 
alter the constants of the torpedo in air since it is 
covered completely by the stabilizer. 

The British aircraft torpedo is usually launched 
with the M.A.T. lA^ stabilizer which has an area in 
the pitching plane of about 1,200 sq in. (60 in. by 
20 in.). With this relatively large stabilizer, the con- 
stants obtained by wind tunnel measurements are 
Cm / a — —4.53 or about three times the value of 
the Mark 2-1 with drag rings. In addition, we may 
expect a correspondingly large Cl/ol since most of 
the restoring moment comes from the lift at the tail. 
Cm/\P = —1.31 or about 1.45 times the value of the 
Mark 2-1 with drag ring. These larger values may be 
considered as due primarily to the larger area of the 
M.A.T. IV stabilizer. 

It is interesting to compare the constants of the 
Mark 13 torpedo in air with its underwater charac- 
teristics (where the stabilizer and drag ring have 
been removed). Thus it has been found in water and 
wind tunnel tests without propellers that for zero 
elevator or rudder angle the Mark 13 torpedo with 
shroud ring (using the density of water) 

Cm/ a = 0.69 (destabilizing), 

Ck = 0.46, 

Cf = 1.16, 

CL/a = 2.24, 

Cd = 0.13. 

It is noticed that the Cm/ol and Ck produced by 
the Mark 2-1 stabilizer with a drag ring are quite 
large. In fact, in the air, due to the stabilizer, Cm/ol 
is negative, or the center of pressure of the lift forces 
is aft of the center of gravity, producing a restoring- 
moment, while underwater the center of pressure of 
the lift forces is forward of the center of gravity. 


5.5 RELATIONSHIP OF QUANTITIES 
RELATIVE TO AIR AND 
TO GROUND 

The primary reason for studying the air flight of 
the torpedo is to permit a prediction of the water- 
entry conditions from a knowledge of the release 
conditions. The condition of the torpedo at entry 
will determine its behavior in the entry stage and in 
the initial underwater trajectory. However, and this 


46 


AIR FLIGHT OF A TORPEDO 


distinction is the essence of this part, the underwater 
behavior of the torpedo will depend on the entry condi- 
tion relative to the water. Up to this point all the 
quantities which have been discussed, such as veloci- 
ties, distances, pitch angles, etc., have been relative 
to the air. 

5.5.1 Ouantitative Considerations 

The reason why the quantities used were relative 
to the air is that the lift and drag forces and the re- 
storing moments acting on the torpedo in a fluid 
(air or water) are determined by the velocity relative 
to the fluid. Thus all quantities so far derived are as 
seen by an observer fixed in the air, and all quantities 
used are relative to air. 

This fact does not alter any of the results obtained. 
However, since the underwater behavior depends on 
the entry conditions as an observer on the ground 
(or water) sees them and since the range relative to 
ground is an important tactical consideration, we 
must examine the results to see what additions must 
be made in order to convert quantities relative to air 
to quantities relative to ground. 

First, it should be noted that, if no wind is blowing, 
that is, if the air has zero velocit}^ relative to ground, 
there is no distinction between quantities relative to 
air and relative to ground. However, if there is, for 
example, a 20 knot head wind blowing, the velocity 
of the torpedo relative to the surrounding air is 20 
knots greater than the velocity relative to ground. 
Similarly, if there is a 20 knot tail wind blowing, the 
velocity of the torpedo relative to air will be 20 knots 
less than its velocity relative to ground. The prob- 
lem reduces itself to examining the effect of the wind 
on the results previously obtained. 

In general, we may transform all quantities derived 
from quantities relative to air to quantities relative 
to ground by the vector relation 

V,, = ^ V,, . (28) 

'Vtg = velocity of torpedo relative to ground, 

Vta = velocity of torpedo relative to air, 

Vaa = velocity of air relative to ground, or the wind. 

Since equation (28) is a vector equation, it is 
equally true for each of the components of the ve- 
locity, i, y, and z. For example. 


where .tag = component of the wind velocity in the 
X direction. 

To consider the effect of the wind, we shall con- 
sider separately its effect on the trajectory and on the 
pitching and yawing motions. It will be assumed that 
the wind remains constant through the drop. Although 
this is rarely the case, the effect produced by varying 
winds is complicated and small and will be neglected. 
The wind velocity used will always be the velocity 
at release. 

5.5.2 Effect of the \Vind on the Trajectory 

Let the wind velocity at release have the three 
components .f«p, yag, tag. 

1. X Direction. As has been pointed out from equa- 
tion (28), 

Xtg = Xta + i'ag (29) 

SO that 

•t tg ‘t ta “b agt j 

where Xtg will be the horizontal range relative to 
ground that the torpedo will traverse in its air flight. 

From these considerations it is seen that with a 
tail wind component the speed of the torpedo or 
plane at release relative to air is less than the speed 
relative to ground, while for a head wind the speed 
relative to air is greater than the speed relative to 
ground. In addition, the amount by which the range 
relative to ground is greater than the range relative 
to air is given by .iagt. 

2. y Direction. If y or y relative to air are to be 
different from y or y relative to ground, then there 
must be a vertical component of the wind. There is 
not much information on this point, but the trans- 
formation is quite simple: 

ijtg — ijta T" dag j 

where dag is positive for a wind blowing vertically 
downward toward the earth. 

3. z Direction. In Section 5.1 the x, y, and z axes 
were chosen so that relative to air at release i = 0. 
As a result, to the finst approximation, the trajectory 
of the torpedo in air was in the vertical {x-y) plane. 
This would mean that at release the torpedo would 
not experience a cross velocity relative to the sur- 
rounding air. Consequently, 


OL ANTITIES RELATIVE TO AIR AND TO GROUND 


47 


or relative to ground the torpedo will be drifting 
perpendicular to its axis at a rate which is the magni- 
tude of the cross wind at release. This is as expected 
for, if the torpedo is not to experience a cross wind, 
it must drift with this wind at release so that the 
relative velocity is zero. With a cross wind the tor- 
pedo will fall in a vertical plane relative to air (ac- 
cording to the way an observer on the torpedo would 
feel the relative air), while relative to ground the 
torpedo would be drifting in the horizontal plane. 
The distance the torpedo will drift relative to ground 
• 


Thus, for a given release velocity relative to air and 
with a tail wind, the trajectoiy angle relative to 
ground {dtg) is smaller than with a head wind. 

Figure 23 illustrates these results. This figure 
shows the trajectories relative to ground that result 
from horizontal release for a constant velocity rela- 
tive to air of 200 knots with no wind, a 35-knot head 
wind, and a 35-knot tail wind. The three trajectories 
are different, even though they are drawn for the 
same relea.se velocity relative to air because, for the 
torpedo to be released at a constant velocity relative 
to air with a tail wind, its velocity relative to ground 



CURVE 

WIND 

GROUND RELEASE 
SPEED (KNOTS) 

A 

35 KNOTS -TAIL 

235 

B 

0 KNOTS 

200 

C 

35 KNOTS - HEAD 

165 


Figure 23. Trajectories, as seen from ground, for various winds. Constant release speed (relative to air) = 200 knots. 


4. 6 Trajectory Angle. The wind will cause the tra- 
jectory angle relative to ground to be different from 
that relative to air. 

A good approximation for dtg at any time during the 
drop is obtained by retaining the first term in a 
Tavlor series, thus 

Utg = = tan (da + Ad) . 

^ta 4 “ ^ag 

Then 

sin 2e, ^ , 
io 

and 

e„ = 0a - i sin 20„ ^ . (30) 

io 

da is given according to Section 5.1.3 by 

tan da — cq 4“ Cit 4“ ^ 2 ^" 4“ * * ■ . 


must be larger than the release velocity relative to 
air; with a head wind the release velocit}^ relative 
to ground will be less than the release velocity rela- 
tive to air. Therefore, relative to ground, for con- 
stant release velocity relative to air, a tail wind tends 
to produce a smaller trajectoiy angle and a head 
wind a larger trajectory angle. 

5.5.3 Effect of ind on Angular Motion 
Pitch Angle 

The presence of a wind component in the direction 
of motion will also cause the pitch angle relative to 
ground, ag, to differ from the pitch angle relative 
to air, aa. 

To find the relation between ag and aa it is to be 
remembered that 

ag = angle of axis of torpedo, dg , 
aa = angle of axis of torpedo, da . 


48 


AIR FLIGHT OF A TORPEDO 


Consequently, 

ag = aa {da — dg) . (31) 

The meaning of this relation is readily understood. 
The torpedo axis is at a certain angle with the hori- 
zontal. The pitch angle is the angle between the axis 
and the tangent to the trajectory. Consequently, the 
difference between ag and is the difference between 
the two trajectory angles. 


that computed from the release velocity relative to 
the air. However, only the correction to the vacuum 
trajectory will be changed, and a computation using 
the time average of the wind in which the torpedo is 
moving will give a close approximation to the exact 
result. In addition, the torpedo oscillation tends to 
damp out until its axis lies along the local air trajec- 
tory. Since this is changing differently from the way 
it changes with a constant wind, the mean position 
will be a little different. The primary effect of a var- 



YAW 

ANGLE 


A -TORPEDO TRAJECTORY RELATIVE TO GROUND WITH NO WIND. 

B - TORPEDO TRAJECTORY RELATIVE TO GROUND WITH 20 KNOT 
WIND IN DIRECTION OF ARROW. 

AIRPLANE DRIFTS WITH THE WIND AT RELEASE SO THAT 
GYROSCOPE IS AIMING IN DIRECTION OF TARGET. 

Figure 24. Trajectories relative to ground in horizontal plane at 200 knot release velocity. 


From (30) it is therefore clear that 

sin 26a — . (32) 

i’o 

Thus it may he concluded that, in general, a tail wind 
tends to produce a nose-down pitch relative to ground 
and a head wind tends to produce a nose-up pitch rela- 
tive to ground. 

This is illustrated in Figure 23 where we have the 
same air trajectory with a 35-knot tail wind, no wind, 
and a 35-knot head wind. In this figure, for sim- 
plicity, the pitch oscillations and mean pitch angle 
have been taken as zero. 

Since the pitch of the torpedo ag is an important 
quantity and the entry behavior is sensitive to small 
changes in this angle, it must be recognized that the 
neglect of the variation of the wind with altitude 
may not be completely satisfactory in this connec- 
tion. The angle da near the ground will not be exactly 


iable wind is on and hence ag. The variation of the 
wind magnitude does not affect the solution of the 
homogeneous equation (20) but only alters the par- 
ticular solution given in equation (21) by altering the 
value of d. Taking this into account and using equa- 
tion (30), one can easily compute aa if the type of 
variation of the wind is known, and, knowing the 
wind at the water level, ag at entry (which we shall 
call ae) is then determined. However, for many pur- 
poses it appears that the use of a suitable mean wind 
might give a satisfactory approximation. 

Yaw Angle 

For a constant cross wind \pg will differ from xpa- By 
analogous reasoning to that used to determine ag, 

I'v = I' a {ua — Ug) • (33) 

For constant cross wind Ma = 0 and equation (32) 




QUANTITIES RELATIVE TO AIR AND TO GROUND 


49 


indicates that a xoind from the left will tend to produce 
a yaw angle to the left relative to ground and a wind from 
the right, a right yaw angle relative to ground. 

This may be understood by the fact that for a con- 
stant cross wind, neglecting the yaw oscillations, the 
torpedo will have zero yaw (at release the pilot tries 
to maintain zero yaw) so that the torpedo is drifting 
with the cross wind. Consequently, if a wind is blow- 
ing from starboard so that the torpedo is drifting to 
port and pointing straight ahead, the result is a star- 
board yaw angle. This is shown for x = 338 ft per 
sec and Zag = 55 ft per sec in Figure 24. 


5.5.4 Summary 

In this part we have examined the effect of wind in 
altering the trajectory, trajectory angle, pitch and 
yaw angles relative to air and relative to ground. 
Given the release conditions of the torpedo and the 
wind, the condition of the torpedo relative to air and 
relative to ground at entry may be predicted. Be- 
cause of the sensitivity of the entry behavior to the 
exact pitch and yaw angles, these effects of the wind 
are of great importance in prescribing the conditions 
for torpedo launching. 


WATER ENTRY 


T he study^ of the air flight was concerned with 
predicting the conditions of the torpedo at im- 
pact from a knowledge of the release conditions. In 
the study of the water entry, it is desired to predict, 
from a knowledge of the entry conditions, the under- 
water behavior of the torpedo from the time of im- 
pact with the water to the time it settles down to its 
normal run. 

The entry conditions of a torpedo may be described 
b}^ the values of the following quantities at the in- 
stant of water impact. 

1. Velocity, IT, 

2. Trajectory angle, 6e, 

3. Pitch angle, ae, 

4. Pitching angular velocity, de, 

5. Yaw angle, \pe^ 

6. Yawing angular velocity, \pe, 

7. Roll angle, 4)e, 

8. Rolling angular velocity, 0^. 

In the subsequent discussion, the effect of these 
quantities on the initial underwater trajectory will be 
traced through the various phases of the trajectory. 
In addition, the effect of physical parameters of the 
torpedo, such as head shape, moment of inertia, mass, 
length, diameter, position of the center of gravity, 
and center of buoyancy, and the tail structure, will 
be considered. 

The phases into which the initial underwater tra- 
jectoiy will be divided for the subsequent discussion 
are 

I. Impact (elastic wave in the water), 

2. Flow forming (head is being submerged), 

3. Motion in the cavity before tail slap, 

4. Alotion in the cavity after tail slap until cavity 
closure and during the subsecpient cavity dissi- 
pation, 

5. The subsequent noncavitating motion of the 
torpedo. 

Although the discussion will be presented in general 
terms and will be specialized only for illustration and 
verification of the ideas presented, it must be re- 
meml)ered that most of the available information is 
based on studies of the Mark 13 torpedo. Most of the 
illustrations will be based on tliis torpedo and the 
points of view of the theory have largely developed 
around it. The Mark 13 torpedo head consists 
roughly of a hemisphere followed by a finely tapered 



Figure 1. Mark 13 torpedo warhead — practically a 
hemisphere on a finely tapered cone. 


cone and is illustrated in Figure 1. Radical depar- 
tures from this shape may alter the relative impor- 
tance of various features of the analysis. 

6.1 IMPACT STAGE (ELASTIC WAVE) 

6.1.1 Pressure at Point of Impact 

Except in the very extreme case in which the point 
of a pointed head makes the first contact with the 
water surface, the impact pressure exerted on the 
head, as calculated for a ‘‘perfect fluid” by potential 
hydrodynamic theory, is infinite since a mass of water 
is given finite kinetic energy in zero time. However, 
since water is compressible, it cannot sustain an in- 
finite pressure, and the pressure on the head will be 
limited by the compressibility of the fluid. 

When the torpedo strikes the water surface, the 
part of the nose that normally makes first contact is 
essentially flat so that a compression wave is gener- 
ated, and the pressure behind this wave may be 
easily calculated. 

The bulk modulus for water is given by 

E = pc- , 

where p is the density of water, 2 slugs per cu ft, and 
c is the speed of sound in water, 4,800 ft per sec. 

Consider a cylindrical element of water of length 
cAt, of unit cross-sectional area and with its axis 
normal to the surface of the water at the point of 
contact. This cylinder, on impact of the torpedo, 
undergoes a compression of magnitude VnrAt, where 
Vnr is the normal velocity of the torpedo relative to 
the surface. The strain (compression) of the water is 
then 

I" At V 

I nrLAi y jiy 

cAt c 


IMPACT STAGE (ELASTIC WAVE) 


51 


so that the pressure on the nose of the torpedo at the 
point of impact, which is the pressure behind the 
elastic wave, is 

FV 

\ nr 

P — — pC I nr • 

C 


of contact of the torpedo, air, and water, recedes to 
diminish to the speed of sound c. The factor of two 
arises since the wave is reflected from the point 
where the velocity of the contact point is just equal 
to c. From Figure 2 it is readily seen that the velocity 
of the point where air, water, and the torpedo meet is 


For a torpedo entering the water at a trajectory 
angle 6e, 


TT = T'esin^e 


so that, if the water surface is initially at rest. 


Ve sin (6 + (3) CSC (3 . 

Consequently, this point of contact is moving faster 
than c until the surface of the torpedo makes an 
angle (3 such that 


f n ^ nr 

and 

p = pcT^e sin lb per sq ft. (1) 

For the ]\Iark 13 torpedo with de = 20°, and 
Ve = oOO ft per sec, p = 11,400 lb per sq in., which 
is many times the stagnation pressure. 

6.1.2 Time Duration of Pressure at Point of 
Impact and Area over Which It Acts 

We now inquire into the time during which this 
pressure at the point of impact may be regarded as 
constant and the area of the nose over which it acts. 
At impact, the point where the torpedo, water, and 
air meet usually recedes from the point of initial 
contact with a velocity greater than c, the velocity 
of sound in the water. This is because of the near 
flatness of the impact surfaces. As a result, the elastic 
wave that is generated at impact, and which pro- 
duces the large pressure, cannot escape. At impact, 
the rate at which this point recedes is infinite. How- 
ever, unless the head is really flat, this rate diminishes 
until its value is less than c, and the wave can then 
escape. The peak pressure persists during the time in 
which the pressure wave cannot escape, and it begins 
to damp out thereafter. The pressure wave begins to 
decay after a time which is roughly ecpial to twice 
the time required for the velocity at which the point 

TORPEDO WATER 



F 


sin d 


c sin {de + jS) 

The value of corresponding to this condition 
depends on Ve and de only and is independent of the 
shape of the head. Generally d is quite small so that 


Then 


or 


sin d = tan (3 = (3 


Ve 




C sin de -j- /3 cos de 


■ sm de 

c 

Y 

1 — — cos de 
C 




For the Mark 13 torpedo, which has a hemispher- 
ical nose of radius r, since /3 is small 


sin d = tan (3 = ^ 


X 


* 


r 


X* is the horizontal distance from the point of impact 
to the place where the contact point (air, water, and 
the torpedo) travels at the velocity c. 

Then, since for most torpedo launchings 

Y 

— cos < < 1 , 
c 


-V.* 


Ve sin der 
c 



Similarly, for the spherical head it can be seen from 
the figure that the time necessary for the point at a 
distance .t* ahead of the point of initial contact to 


Figure 2. Diagram of torpedo nose entering the water. 


52 


WATER ENTRY 


reach the undisturbed surface of the water is approx- 
imately 

^ r(l - cos^) ^ X* 

T^esin^e 2c 

During a time of about twice this length the pressure 
continues at near its peak value. Hence for the Mark 
13 torpedo the pressure begins to decrease after the 
time 

2t* = — sin deV . (3) 

c- 

With Ve = 500 ft per sec and Be = 20° this time 
amounts to about 6.4 microseconds. 

The area over which the pressure acts is given 
roughly by 

A = 7rX“ 

as long as x < x*. The maximum value of this quan- 
tity, A* = TTX*^ for the Mark 13 torpedo at 500 ft per 
sec and de = 20°, is about 0.58 sq in. 

A similar approximate calculation can be carried 
through for any shape of nose, and an estimate can 
be obtained of the time and area over which the 
maximum pressure acts. The more pointed the nose, 
the faster the increase in (3 and hence the smaller t* 
and X*. The extreme roughness of the estimate, how- 
ever, must always be kept in mind. The above calcu- 
lation of X* really only applies to the forward direc- 
tion, and the values of the rear and to the sides will 
certainly be different. 

6.1.3 Effect of Elastic Pressure Wave 

With the physical description of the phenomena at 
impact, we may estimate the effect the elastic wave 
pressure on the nose has on the damage sustained by 
the torpedo and the influence of this pressure on the 
trajectory and underwater behavior of the torpedo. 

From ecpiation (1) we know the magnitude of pres- 
sure the head of the torpedo must withstand at the 
point of impact. This is not particularly significant 
for, as will be shown later, the nose must sustain even 
greater pressures at later stages in the entry. 

Since the time for which this constant pressure 
wave persists is very small and the area of the nose 
over which it acts is also small, we can expect both 
the force and the impulse due to this pressure to be 
small. 

The force in the longitudinal direction is given ])y 
Fi = pA* sin de = TrpcVeSm‘^ de{x*y 


and in the transverse direction 

Ft = pA"^ cos Be = TpcVc sin Be cos . 

For the Mark 13 torpedo with Ve = 500 ft per sec. 
Be = 20°, the numerical values are 

Fi = 2.26 X 10-' lb , 

Fo = 6.22 X 10' lb , 

and the impulse is given by 

t 

Fdt < Fi 

so that the longitudinal impulse for the Mark 13 is 

1 1 < 1.45 X 10~'Mbsec , 

1 2 < 3.98 X 10““ lb sec . 

Consequently, we can say that the resulting change 
in the linear or transverse velocities due to the elastic 
pressure wave is entirely negligible. 

However, as will be shown shortly, in later stages 
of submergence of the nose, large compression wave 
pressures and smaller hydrodynamic pressures acting 
for a longer time are produced which have marked 
effects on both damage and the trajectory. 

6.1.4 Effect of Drag Ring and Nose Cap 

When the Mark 13 torpedo is fitted with the Mark 
1 drag ring (pickle barrel), we cannot expect the 
pressures at the point of impact to be predicted by 
(1), but we expect them to be less than these values. 
The reason for this is that, when the drag ring is on 
the nose of the torpedo, it strikes the water first and 
has two effects: 

1. The drag ring starts the water moving before 
the nose of the torpedo comes in contact with the 
water, with the result that the relative velocity of 
the torpedo to the water is diminished, F,- < W. 

2. The drag ring serves as a “cushion” so that 
the pressure on the nose of the torpedo would be 
governed l)y the compressibility of wood which is 
greater than that of water. 

In some launchings the Mark 13 torpedo is fitted 
with a nose cap over the drag ring (in order to reduce 
the drag while the torpedo is being carried by the 
airplane). If the torpedo entered the water with the 
nose cap, we should expect a similar reduction in the 
pressure at impact due to its presence for the same 



FLOW-FORMING STAGE 


53 





reasons as the reduction due to the drag ring. These 
effects and the experimental verification of the pre- 
vious anabasis have been observed at the California 
Institute of Technology Torpedo Launching Range 
[CIT-TLR]. 

6.1.5 Summary of Elastic Impact Stage 

At impact the water is compressed, forming a pres- 
sure wave. The magnitude of this pressure is given 
by (1) and is proportional to Ve and sin For the 
Mark 13 torpedo with TA = 500 ft per sec, 0e = 20°, 
we have a pressure p = 11,400 lb per sq in. acting 
for a time t = 12.9 X lO"*^ seconds; the area over 
which this acts at this time is 0.38 sq in. The effect 
of the pressure wave on the damage is not important 

ntfCKR 


since the nose must sustain even greater pressure as 
will be seen later, and its effect on the trajectory is 
negligible. Furthermore, the pressure at the point of 
impact is considerably diminished by the drag ring 
or a nose cap. In fact, this entire stage is of such short 
duration that it is usually negligible. 

6.2 FLOW-FORMING STAGE 

This stage of the motion lasts until the head of the 
torpedo is submerged so that the flow breaks away 
from the top side of the nose. During this stage the 
torpedo undergoes a change in its longitudinal ve- 
locity and a change in its angular velocity. The de- 
tails of these changes have not been observed, except 
in models, and, since they take place in a very short 







54 


WATER ENTRY 


time, they will be treated as impulsive changes. The 
impulsive change in angular velocity in the vertical 
plane is called the whip. It is the magnitude and di- 
rection of the whip that is the important consequence 
of this stage of the motion. The phenomena occurring 
are understood qualitatively, and some quantitative 
results have been obtained experimentally ; however, 
it is in this stage that a quantitative theoretical ap- 
proach is most lacking. 

This stage is also of importance from the point of 
view of shell damage since there is a high peak de- 
celeration during the nose submergence. 

The most important physical characteristic of the 
torpedo during this stage of the motion is its nose 
shape. It is the shape of the nose that primarily in- 
fluences the whip at entry, and this in turn has a con- 
trolling effect on the subsequent underwater tra- 
jectory. 

6.2.1 Duration of the Flow-Forming Stage 

When the torpedo nose enters, the water will flow 
around it until separation occurs both from the top 
and from the bottom. One can say briefly that the 
flow will separate from the nose at any point where 
the total pressure on the nose is roughly zero or at any 
discontinuity on the surface of the nose for at a dis- 
continuity the theoretical pressure becomes nega- 
tively infinite and the flow must separate. The point 
on the nose where the total pressure becomes zero 
depends on the pressure in the air or vapor near the 
torpedo head. Since at the time we are concerned with 
flow separation the torpedo head is still in the air, 
the pressure around the head is roughly atmospheric 
pressure or perhaps somewhat less than atmospheric 
pressure. 

Consequently, it is clear that the time duration of 
the flow-forming stage is intimately connected with 
the nose shape. In order to calculate the point at 
which the flow breaks away for ‘‘smooth” heads like 
spheres and ogives, the pressure distribution must 
be known. This point is, however, still somewhat 
obscure. Nevertheless, for noses with discontinuities, 
such as cones, flat heads, truncated cones, noses with 
kopfrings and spigots, examples of which are illus- 
trated in Figure 3, the flow will separate at the dis- 
continuity of the nose. Thus it is possible to calculate 

^ There may be some circulation of the air in the rej 2 :ion near 
the place where the head, water, and air meet, which could 
reduce the pressure. It is estimated by the Morris Dam Groiij), 
which works on models, that this reduction of pressure is 
about to 1 atmosi)here. 


roughly the duration of the flow-forming phase of the 
motion for various shaped noses. 

Let rjs be the angle at which the flow separates from 
the upper side of the nose. This angle is constant for 
all hemispherical heads, and different estimates of its 
magnitude vary from 42° to 65°. If r is the radius of 
the hemispherical nose, then the time to separation of 
the flow from the upper side of the nose is given 
roughly by 

^ ^ r[ I + sin {r]s — de)] 

Ve sin de 

This is illustrated in Figure 3. 

A rough picture of flow separation on a one-inch 
model of the CIT Steel Dummy is illustrated in 
Figure 4. In this figure the one-inch model is seen 



Figure 4. Flow separation on 1-iii. model of Steel 
Dummy at end of flow-forming stage (de = 19°). 


just after the end of the flow-forming stage. This is 
not necessarily an exact model of the full-scale tor- 
pedo behavior but it illustrates the concept of flow 
separation and flow-forming stage. 

If T 7 s = 55°, for de = 20°, Ve = 500 ft per sec, 
t = 8.6 X 10“^ sec = 8.6 msec. 

For heads of other shapes an approximate time to 
flow separation can be estimated along lines indicated 
in Figure 3. 

It should be remembered that, while the flow sepa- 
rates from the nose at the various points discussed 



FLOW-FORMING STAGE 


55 


above, if the cavity angle of the flow at separation is 
not great enough, it may easily reform on a part of 
the body further aft. This has been found with spigots 
in some experiments on nose shapes with rockets and 
on other projectiles like the Shark when yawed. Thus 
a second kopfring was put on the Shark head since 
the flow which broke away from the first kopfring 
when the projectile was pitched or yawed reformed 
on the body. The flow broke away from the second 
kopfring when the Shark projectile was yawed and 
then cleared the entire body. 

6.2.2 Pressures Acting on the Torpedo Nose 

During the time of submergence of the nose of the 
torpedo large pressures are acting on it. At the point 
of impact the pressure on the nose is approximately 
given by equation (1). As the torpedo enters the 
water and before the Avater has started to move, 
there will be points on the torpedo ahead of the point 




INITIAL PART OF 
FLOW-FORMING PHASE 

Figure 5. Schematic illustration of elastic wave and 
initial part of flow-forming stages. 

of impact at which the relative velocity of the tor- 
pedo to the water will be greater than at impact (see 
Figure 5) . Thus, at a point on the nose where the sur- 
face of the nose makes an angle 13 with the water 
surface, the velocity relative to the water is 


Ve sin (de + (3) — Vrv , (4) 

where TT = velocity of the water. Consequently, if 
TT = 0, the pressure at this point on the nose is given 

by 

P = pcVe sin (de + (3) . (5) 

Since sin (de (3) > sin de for points forward of the 
point of impact, the pressure at points forward of 
the point of impact is expected to be greater than 
at the point of impact. Similarly, it is expected that 
the pressure aft of the point of impact should be less 
since the relative velocity is smaller. Presumably 
these peak pressures begin to attenuate immediately, 
somewhat as illustrated in Figure 6. 



The CIT-TLR uses hydropressure plugs in order 
to record peak pressures at various points on the nose 
of the Mark 13 torpedo, as illustrated in Figure 7. 
About twelve plugs are inserted so that the distribu- 
tion of the peak pressures over the nose is recorded. 
Examples of the results obtained are given in Figure 
8. From this figure the effect of the drag ring and nose 
cap in reducing the elastic wave pressure is apparent. 
From equation (5) Ave might expect to predict the 
pressure distribution over the head except that the 
relative velocity of the torpedo to the surrounding 
Avater is not knoAvn. It is to be expected that, as the 
nose is submerged and momentum is imparted to the 
Avater, the Avater Avill floAv and diminish the velocity 
of the torpedo relative to the surrounding Avater. 
HoAvever, in the neighborhood of the point of im- 
pact, the Avater has not had a chance to floAv so that 
the recorded pressures should be roughly those pre- 


56 


WATER ENTRY 



F IGURE 7. Mark 13 nose fitted with pressure plugs. 

dieted by equation (1), which is equation (5) with 
jS = 0. In Figures 9, 10, and 11 the observed pressures 
at the point of impact (JS = 0), at the point 16° for- 
ward of the point of impact (jS = 16°) and 16° aft of 
the impact point (/3 = — 16°), are compared with the 
pressures predicted by equation (5). On the whole, 
considering the experimental errors and the very 
crude nature of the theory, the agreement is good. 

Forward of the position which is 16° forward of the 
point of impact, we notice from Figure 8 that the 
peak pressures are significantly less than indicated by 
the estimate. Here the water has had time to acquire 
a velocity so that Vw > 0, and the relative velocity of 
the torpedo to the water is less than Ve. We do not 
know the magnitude of Vic, but we do know that the 
calculated pressures must be too high. 

As has been mentioned, these compression wave 
pressures attenuate very rapidly so that they only 
exist for a very short time. However, when they have 
damped out, the hydrodynamic pressures remain 
about a body which is decelerating in a fluid, but 
around which the flow is still forming so that the 
water has not yet separated from the top of the nose 
of the torpedo. These pressures probably vary as V-. 
They are smaller in magnitude than the compression 
wave pressures, but on the other hand they act over a 
larger area and exist for a longer time. 

Various theoretical attempts have been made to 
determine the distribution of these hydrodynamic, 
V^, pressures over the nose of the torpedo for simple 



MARK 13 NOSE 
FITTED WITH MK'I 
DRAG RING 
PLUS NOSE CAP 
2 LAUNCHINGS 


Figure 8. ]\Ia.\imum pressure distributions obtained 
by pressure plugs (A, B, etc.) at CIT-TLR. 


head shapes. So far, only some results for the vertical 
entry of a sphere have been obtained by two different 
theoretical approaches. These results are not directly 
applicable to the problem of oblique entiy, which is 
the practical problem we are concerned with in the 
water entry of torpedoes. These results indicate that 
the pressures on the nose during this stage of entry 
ai'e many times the stagnation pressure. The theoreti- 
cal results obtained for this simplified case are in fair 
agreement with experimental results, but more work 
should be done along these lines. 

C'onsequently, we have no theoretically reliable 



FLOW-FORMING STAGE 


57 



0 100 200 300 400 500 600 700 

ENTRY VELOCITY 


Figure 9. Pressure at point of impact (i3 = 0°) versus 
entry velocity for Mark 13 torpedo; Station F. 



Figure 10. Pressure at point forward of impact 
(/3 = 16°) ; Station E. 



Figure 11. Pressure at point aft of impact (j3 = 
— 16°); Station G. 



Figure 12. Mark 13 nose without drag ring “dim- 
pled” by high-speed entry. 



Figure 13. Mark 13 afterbody buckled (“accordion 
pleated”) by high-speed entry. 


58 


M ATER ENTRY 


quantitative results for the T - pressures on the nose 
of the torpedo for oblique entry, but we do have 
many qualitative results as well as experimentally 
obtained quantitative results. ^lany of these will be 
discussed when the trajectory in the flow-forming 
stage of the motion is described. 

6.2.3 Damage During Flow-Forming Stage 

The damage to the torpedo at entry may be di- 
vided roughly into damage to the shell and damage to 
the internal mechanisms of the torpedo. 

Damage to the Shell 

Damage to the shell of the torpedo is due to the 
high peak pressures which last for a relatively short 
time. In the case of the IMark 13 torpedo, the very 
large pressures predicted by equation (5) have been 
found to “dimple’' the nose of the torpedo. This has 
been observed both at the Newport Torpedo Station 
and the CIT-TLR. In Figure 12 the photograph of 
the nose of a ]\Iark 13 torpedo launching at the CIT- 
TLR in which the nose was “dimpled" is illustrated. 
The point of “dimpling" is about 30° off the axis of 
the nose and occurs close to the point of maximum 
pressure. 

As has been discussed in Chapter 5, the drag ring 
diminishes the peak pressures, as may be seen from 
the pressure plug readings in Figure 8. Consequently, 
this type of nose damage should be considerably 
diminished by the presence of the drag ring. Further, 
it has been found that the peak deceleration of the 
^lark 13 torpedo during this stage of the motion is 
very large and varies roughly as or faster. These 
peak decelerations are only local in nature and have 
little relation to the motion of the torpedo. They are 
measured by accelerometers in the nose and represent 
local deformations. The recorded magnitudes are 
over 1,200^ in a direction transverse to the torpedo 
axis. It is to be expected that the forces causing this 
high deceleration would damage the shell. With the 
drag ring on the nose, this peak deceleration is 
diminished to about 75 per cent its value without 
the drag ring. Thus it was found that, for = 400 ft 
per sec for the CIT Steel Dummy (3/ = 47.2 slugs), 
for 6e = 20°, the accelerations as recorded by in- 
denters in the nose are 


Further verification of these predictions and re- 
sults have been obtained in launchings at the New- 
port Torpedo Station. There it was found that, for 
launchings for which roughly Ve = 450 ft per sec, 
without a drag ring the torpedo nose was “dimpled" 
at entry, and the afterbody was “accordion pleated," 
that is, it was buckled as is expected from a very 
high peak axial deceleration. This last type of dam- 
age has also been observed at the CIT-TLR and is 
illustrated in Figure 13. However, when fitted with a 
drag ring, none of these effects was noticed in the 
entire velocity range at which torpedoes were 
launched at the Newport Torpedo Station (up to 
about Ve = 600 ft per sec). 

Another type of shell damage occurring during this 
stage of motion is the failure of the joints of the tor- 
pedo. Due to the nose-up angular velocity received 
during this stage of the motion, large stresses are ex- 
perienced at the joints of the torpedo shell. This 
failure has been observed at the CIT-TLR and is 
more marked the more the pitch is nose-up at entry 
since this increases the nose-up whip. 

Component Damage 

Component damage to the torpedo is rareh^ pro- 
duced by this sudden peak deceleration except per- 
haps on some types of exploder mechanism in the 
nose. The behavior of the components when subject 
to a sudden short peak deceleration is as though they 
had suddenh' received the impulsive velocity change 
sustained by the torpedo. Thus it may be proved 
that the acceleration sustained b}^ the components of 
the torpedo is proportional to the impulsive change 
in velocity. The main source of component damage is 
the steady drag deceleration which will be discussed 
further under Torpedo Damage in Section 6.3. 

6.2.4 Impulsive Axial Velocity Change 

General Considerations 

Since, during this relatively brief stage of the mo- 
tion, momentum is conserved, we ma}^ say that 

MYe = (3/ + m) {Ve - AT) , 

where ??? represents some suitable mass of water 
which is being accelerated. This may be called the 
induced mass. Consequently 


Plain no.se oOog, 

Drag ring 405^, 

Drag ring and nose cap 340^. 




( 6 ) 


FLOW-FORMING STAGE 


50 


or the axial loss in velocity is proportional to the 
entry velocity. The value of the quantity' m, the mass 
of water being accelerated, is obscure. 

Several suggestions have been made for the value 
of w to be associated with the entry of a sphere in 
water. 

1. It has been proposed that the value of m be 
taken as half the induced mass corresponding to the 
potential flow about a sphere whose radius is given by 
the radius of the circle of intersection of the water 
surface and the entering sphere, that is, l/.S^pH, 
where r is the radius of the circle. (Of course, r is a 
function of the depth of penetration of the sphere.) 

2. It has also been suggested that the value of m be 
taken as the induced mass corresponding to the po- 
tential flow about a disk whose radius is the radius of 
the circle of intersection of the sphere and the water, 

In addition, in one theoretical attempt at the cal- 
culation of the vertical entry of a sphere in water the 
value of m has been obtained by replacing part of the 
sphere below surface by a lens with some refine- 
ments and using for the virtual mass the results ob- 
tained by hydrodynamic theory for the induced mass 
of a lens. The results were not too different from the 
results obtained using m associated with a sphere as 
above. However, no theoretically reliable estimates 
of the quantity in are at hand for oblique entry, so 
we can only sa}^ that it is expected that V will be 
proportional to Te. 

In order to discuss the magnitude of AT, we must 
decide on what distance of submergence of the head 
we are considering in the measurement of AT. The 
distance will be taken to equal the distance corre- 
sponding to flow separation from the upper side of 
the nose, which is defined as the duration of the flow- 
forming stage. It would probably be desirable to be 
able to attribute a certain amount of this change in 
axial velocity during the entire flow-forming stage to 
the compression-wave pressures which we found to 
varv as IT and the balance to the hvdrodvnamic 
pressures which vary as However, there is not 
sufficient evidence, even for the Alark 13 torpedo, to 
be able to make this kind of division. Both types of 
forces usually exist at the same point on the nose at 
different times. Furthermore, for both types of forces 
the net impulse varies directlv as the velocitv. Thus: 

1. Forces varying as Ve, since the}' are compres- 
sion-wave forces and attenuate very rapidly, may be 
considered as acting for a very short time which is 


constant independent of Ve. The impulse is therefore 
proportional to Ve. 

2. The hydrodynamic forces vary as TV, and the 
time for which they act is roughly the time duration 
of the flow-forming phase of the motion. This varies 
as \/Ve. Consequently, for this type of force the 
impulse also varies as Ve. 

As a result, from records on hand which give the 
axial velocity as a function of time for time intervals 
of a millisecond we cannot di.stinguish the effects of 
these two types of forces. 

Magnitude of AT^ 

Because of the inadequacy of the available theo- 
retical estimates of m, the magnitude of Al^ can be 
obtained only from direct observation. Unfortu- 
nately, there are experimental results for only one 
shape nose, namely, the Alark 13 hemisphere. 

For the Mark 13 it is found that, during the flow- 
forming stage, approximately Al^ = 0.034TT. AT^ is 
proportional to IT, and the time duration of this 
stage of the motion is proportional to l/TC- Since 
AT = (a/2) A/, where a is the mean acceleration during 
this phase of the motion, we can say that a a TV- 
At the CIT-TLR, accelerometer records obtained in- 
dicate this fact very clearly. 

In general, as the dimensions of the nose increase, 
the time duration of the flow-forming phase increases 
so that Al^ during the flow-forming stage will increase 
with the dimensions of the nose. In addition, it is 
expected that, as the cross-sectional area of the nose 
increases, the area over which the hydrodynamic 
forces act will increase with the result that AT will 
increase roughly as the cube of the linear dimensions. 

Xot much experimental information has been ob- 
tained for other shaped noses. However, it is probably 
true that AT increases for blunter noses and decreases 
for finer noses. For cones up to moderately large cone 
angles, for ogives, and for spigots, Al^ is probably 
less than a hemisphere; while for truncated cones 
perpendicular to the axis of the torpedo with a large 
flat area, for flat noses, for cone noses with large cone 
angles, and for kopfring noses, ATMs probably larger 
than a hemisphere. 

Since, for a given set of entry conditions a certain 
amount of axial momentum is imparted to the tor- 
pedo, for AT" < < Ve, and m < < M, d/AT" = con- 
stant. As a result, AT^ cc 1/M, or the axial change in 
velocity is inversely proportional to the mass of the 
torpedo. 


60 


WATER ENTRY 


It should be noted that the drag ring on the Mark 
13 torpedo does not alter AT", even though it dimin- 
ishes the peak pressures as obtained by the pressure 
plugs and the peak local decelerations in the nose. 
This is perhaps further evidence for indicating the 
predominant effect of the T"- forces in producing AT^ 
since the drag ring affects the peak values of the pres- 
sure and deceleration, but probably not the steady T"“ 
pressures. 

6.2.5 Change in Angular Velocity of the 

Torpedo in the Flow- 
Forming Stage 

The whip at entry, which is the change in angular 
velocity in the vertical plane, is probably the most 
important single quantity in determining the initial 
underwater trajectory of a torpedo. The forces pro- 
ducing the whip are determined primarily by the 
shape of the nose of the torpedo. 

Two effects contribute to the whip. During the 
flow-forming stage, first the lower part of the nose is 
being wetted and then the upper part of the nose. As 
a result, while the lower part is in contact with the 


water and the upper part is still in air, large forces 
will be acting on the lower part of the nose with no 
compensating forces on the upper part. In general, 
this will apply a torque about the center of gravity of 
the torpedo and produce an angular velocity. The 
second cause of a whip is that, if the torpedo enters 
with a pitch angle ae or a yaw angle xpe, the drag 
forces which produce the change in the axial velocity 
will have a moment about the center of gravity of 
the torpedo which is proportional to sin ae and sin xpe 
and consequently result in a whip Aco and a yawing 
angular velocity Ai/^. 

Since the pressure distributions around the nose 
of the torpedo during this stage of the motion are not 
known, the whip cannot be calculated. The most im- 
portant single unsolved theoretical problem con- 
nected with the water entry of torpedoes is a deter- 
mination of the whip at entry and its dependence 
upon the entry conditions. 

Although a completely theoretical approach has 
not been derived, some tendencies and simple rela- 
tions may be obtained. The whip is very closely con- 
nected with the change in the transverse velocity in 
the flow-forming stage. The whip multiplied by the 



Figure 14. 


Typical tail flare records showing refraction due to whip. 





FLOW-FORMING STAGE 


61 


distance from the instantaneous center of rotation to 
the nose of the torpedo is equal in magnitude to the 
change in the transverse velocity of the nose. 

The change in velocity at entry is proportional to 
Ve. Consequently, the whip at entry is proportional 
to Ve. To review briefly the reasoning: for forces 
proportional to V~, the time while the forces are un- 
balanced (acting on the lower side of the head and 
not the upper side) varies as l/T^ so that the impulse 
will vaiy as T"e. Similarly, the part of the whip which 
is proportional to sin xpe = and sin ae = ae. is also 
proportional to the axial change in velocity AT" and 
hence is expected to be proportional to Finally, 
any part of the whip attributable to the elastic wave 
will vary as T"e. 

Detailed discussion of the whip must be based on 
an assumed nose shape. 

Hemisphere 

Whip Change in Angular Velocity. At the CIT- 
TLR, experimental results have been obtained for a 
hemispherical nose. By means of flares on the tail of 
the torpedo records like Figure 14 are obtained, and 
the change in angular velocity is recorded. 

Since the whip is proportional to Ve and is expected 
to be proportional to ae and xpe as may be seen from 
Figure 15, relations of the form 

= a,(e.) + a2{e,)ae , (7) 

(Xose-up pitch and nose-up whip are positive.) 

^ = a,{e.)4^, , (8) 

^ € 

where 1 is the moment of inertia of the torpedo about 
a transverse axis through the center of gravity, are 
expected. 

The coefficient ai is a function of de. There will be 
a whip simply because one side of the nose is ex- 
periencing hydrodynamic pressures and the other 
side is in contact with air. 

By means of flare records on the tail of the torpedo, 
the coefficients ai and 02 were found at the Full-Scale 
Launching Range with Be = 20° for the Steel Dummy 
torpedo. The external shape of the Steel Dummy is 
the same as the Alark 13, but its moment of inertia 
I = 800 slug-ft^ and M = 47.2 slugs, as compared to 
I = 960 slug-ft^ and M = 67.2 slugs for the Mark 13 
torpedo. 


DRAG FORCE 



Figure 15. Forces independent of torpedo axis orien- 
tation; hence rotational moment of drag force is propor- 
tional to ae. 

Thus, for the CIT Steel Dummy, 

ai(20°) = 187 slug-ft degrees , 

02 ( 20 °) = 21.2 slug-ft. 

In obtaining these quantities from the observations 
a correlation coefficient of 0.8 was found, which is an 
indication of the correctness of the expectations. In 
addition, although there is no experimental evidence 
to give Os, since we are dealing with a hemispherical 
nose, it is clear from symmetry that 03 = Ci 2 . 

We now inquire into the values of Oi and 02 for the 
Mark 13 torpedo. It is clear that the angular momen- 
tum transferred to the Steel Dummy will be the 
same as to the Mark 13 so that these coefficients will 
be the same. However, if the angular momentum im- 
parted to both types of torpedoes is the same, then 
the quantities Aa;/Fe and AxpjVe will evidently differ, 
being in the ratio of the respective moments of inertia 
800/ 960. Thus we may say: 

Steel Dummy at Be = 20° 

Aoi/Ve = 0.234 + 0.0265 ae degree per ft , 

and 

AxpjVe = 0.0265 xpe degree per ft. 

Mark 13 torpedo at Be = 20° 

AbijV e = 0.195 + 0.0221 ae degree per ft , 

and 

AxpIVe = 0.0221 xpe degree per ft. 


62 


WATER ENTRY 


To be able to predict the whip at any value of de, 
we have to know the dependence of Oi on de. On this 
point there is no experimental evidence on full-scale 
torpedo launchings since the whip has only been ob- 
tained for full-scale launchings with de = 20°. To find 
this dependence, detailed assumption on the type of 



DISTANCE FROM ENTRY IN FEET 


Figure 16. Pitch angle versus distance from entr}* 
for 1-in. model of Steel Dummy; de = 12’. 

pressure distribution is unnecessary for one needs 
primarily only the dependence of the pre.ssure distri- 
bution on de. If the pressure distribution is indepen- 
dent of de, as appears likely, for a hemispherical nose 
and if the mechanism of the whip is understood cor- 
rectly as being due to an unbalanced pressure acting 
on the underside of the nose, the whip for zero pitch 
should be proportional to the time duration of the 
flow-forming stage, which was found to be propor- 
tional to cot de. A theory which will give the complete 
pressure distribution, and hence the whip, will also 
^neld the dependence of a i on de. We can only say un- 
equivocally that Qi is a decreasing function of de. and 
there is some indication that it decrea.ses like cot de. 


By means of an optical whip recorder the Morris 
Dam Group working on 1-in. models of the CIT Steel 
Dummy recently obtained the whip as a function of 
distance along the trajectory during the flow-forming 
stage for de = 12°, 19°, and 34°. These are illustrated 
in Figures 16, 17, and IS. These whips are only about 



for l-in. model of Steel Dunmiy; de = 19°. 

70 per cent of the values obtained for the full-scale 
dummy at the Torpedo Launching Range; neverthe- 
less, the dependence of these whips on de is interesting 
to note. Reading the whip, which is proportional to 
the slope of these curves at a distance corresponding 
to the end of the flow-forming stage, one finds that 
the whip varies like cot de. This is illustrated in Figure 
19. There are a number of discrepancies between the 
whip of the l-in. model and the whip of the full-scale 
torpedo so that this is not conclusive evidence that 
Oi cc cot de. However, it is a noteworthy experimental 
indication of the dependence of Oi on de. 

There is even less experimental evidence on the be- 
havior of Qo and az as functions of de, but they prob- 
ably both have finite values when de = 90° and in- 
crea.se as de decreases to small angles. 


FLOW FORMING STAGE 


63 


From relation (7) it is clear (with the sample values 
of Qi and Qo) that for a hemispherical nose the nose-up 
whip at entiw' increases for nose-up (flat) pitch and 
decreases for nose-down pitch and also decreases with 
increasing trajectory angle at entiw', 6^. In other 



Figure 18. Pitch angle versus distance from entrj’ 
for 1-in. model of Steel Dummy; 0, = 34°. 


w ords, the torpedo tends to “stub its toe’’ as it enters 
with increasing nose-down pitch. It is also clear that 
a nose-right yaw angle will produce a nose-right 
angular velocity. 

Of course, the quantities Aco and M are not the 
angular velocities, but only the change in the angular 
velocities. In other words, they represent the angular 
velocities at the end of the flow-forming stage less the 
angular velocities at entry, which are usually rela- 
tively small. 

Change in Orientation of the Torpedo Axis during the 
Flow-Forming Stage — Instantaneous Center of Rota- 
tion and Effect on Trajectory. For a hemispherical 
nose all forces act normal to the surface of the nose 
and the re.sultant force passes through the center of 
the nose. Furthermore, 

Ti = distance from the point of application of the 
impulsive forces to the center of gravity; 

To = distance from the center of gravity to the in- 
stantaneous center of rotation; 


k = radius of g>Tation of the torpedo about the 
center of gra\dty; 

then 

rir2 = . 

For the 

CIT Steel Dummy, ri = 4.9 ft , k- = 17 ft- , 

Alark 13 Torpedo, ri = 4.9 ft , k- = 14.3 ft- . 

Consequently, 

CIT Steel Dummy, r -2 = 3.47 ft , 

Alark 13 Torpedo, r-i = 2.92 ft . 

Since the center of gravity is 5.8 ft from the nose, 
the instantaneous centers of rotation are 

CIT Steel Dummy, 9.27 ft aft of the nose, 

Alark 13 Torpedo, 8.72 ft aft of the nose. 



At the Full-Scale Launching Range the whip at 
entry is obtained from flares on the tail of the CIT 
Steel Dummy. If the torpedo receives a whip at 
entry, it is clear that the tail flare images will fall 
below the air flight line. If the whip is Acc radians per 
sec and if the rate at which the flare images fall be- 
low the air trajectory is Air, then it follows that the 


64 


WATER ENTRY 


distance between the tail flare and the center of 
gravity is given by 

, Aw 

a = . 

Aw 


And from Section 6.2.1 


21 sin de 


r [ 1 + sin {r]s — de)] . 


(11) 


For the CIT Steel Dummy, d = 4.3 ft. Consequently, 
the distance from the instantaneous center of rota- 
tion to the nose of the Steel Dummy is (13 — 4.3) ft 
= 8.7 ft, which is in fair agreement with the calcu- 
lated position. 

Since the instantaneous center of rotation is aft of 
the center of gravity, it is evident that during the 
flow-forming stage the center of gravity will ex- 
perience a transverse velocity of magnitude : 

V = r2Aw . 


Consequently, the sum of the changes in the pitch 
angle for a hemispherical head during the flow-form- 
ing stage from equations (9) and (11) is 

(«! + a 2 ae) r r[l -h sin (ry, - ^e)] 

Aa = r2 — . 

I L 2 sin 

The net result is that for the Alark 13 torpedo with 
de = 20°, y]s = 50°, the pitch angle is made more nose- 
down in the flow-forming stage by an amount de- 
pending on ae and given by 


Consequently, the trajectory (the path of the center 
of gravity) will be refracted upward by an amount 
given by 

tan (A«) = ^ , 

^ € 


and, since Ad is very small, then tan (A0) = Ad so that 


T 2 ACO T2{cii T Cl20ie) 

It I 



For ae = 0, using the calculated values of r 2 . 

Ad (Steel Dummy) = —0.81 degree, 

A^ (Alark 13 Torpedo) = —0.57 degree. 

For ae = —2° (2° nose-down). 

Ad (Steel Dummy) = —0.63 degree. 

Ad (Alark 13 torpedo) = —0.44 degree, 
where Ad is negative since d is being diminished. 

Change in Pitch Angle during Flow-Forming Stage. 
Since the trajectory is refracted upward by an 
amount Ad, the pitch angle relative to the trajectory 
will be made more nose-down by the same amount. 
Consequently, due to the refraction of the trajectory 
the pitch angle during the flow-forming stage is made 
more nose-down by an amount given by equation (9) . 
However, during the flow-forming stage, the torpedo 
is receiving a whip. The nature of the rate of increase 
of this whip during the flow-forming stage is not 
known. However, it is fairly safe to assume some sort 
of linear increase in the whip. As a result the pitch 
angle is made more nose-up by an amount 


Aa = ^ AcoAt = 


(ai T a2ae) VeAt 


( 10 ) 


Aae(ae = 0°) = —0.18 degree (nose-down) , 

Aae{ae = — 2°) = —0.14 degree, 


which is a very small amount. However, it is seen to 
be a function of the trajectory angle and can even 
change sign for a small enough trajectory angle at 
entry. 

For the yawing motion the change in yaw angle is 
given by 



as^e _ r[l -f- sin (tjs — de)] 
/ _ 2 sin de 


f 


and is zero for \pe = 0, while for the Alark 13 tor- 
pedo with \pe = 2°, de = 20°, r]s = 50°, Axp = 0.04 de- 
gree. 

Other Shaped Noses 

For a hemispherical nose we have a great deal of 
quantitative evidence concerning the whip; for other 
shaped noses most evidence is qualitative. 

It is not necessarily true that when a torpedo enters 
the water it should receive a nose-up whip for the 
direction of the whip will depend both on the entry 
conditions and on the nose shape. Even a hemispher- 
ical head can have a nose-down whip at entry if the 
pitch angle at entry is large enough nose-down. 

By considering the general qualitative way in 
which the flow forms around a nose, some idea can be 
obtained as to the nature of the whip. A flat nose, for 
example, will be expected to whip nose-down as illus- 
trated in Figure 20, while a sharp ogive will be ex- 
pected to whip nose-up. Presumably the whip for 
almost any nose shape can be approximated by an 


2/ 


FLOW-FORMING STAGE 


65 



Figure 20. Flow-forming stage of fiat nose torpedo 
showing nose-down moments. 


expression of the form (7) but the functions aiide) 
and a 2 {de) can only be determined by experiment or 
a further development of the theory. 

In one case for a 90° cone tangent to the Mark 13 
(hemispherical) nose, with a radius of about 7.29 in. 
at the end of the cone, values of the coefficients a \ and 
^2 were obtained at the CIT-TIJl for de = 20° with 


angle with this cone nose than with a hemisphere. For 
a nose with a spherical cap of large radius, one may 
expect a small whip as may be seen in Figure 21. 



Figure 21. Small moment arm and hence small whip 
of torpedo with spherical cap of large radius. 

Evidence on the Nature of Forces 
Producing the Whip 

It has been mentioned that the whip at entry is ob- 
tained at the Full-Scale Launching Range by means 
of flare records on the tail of the torpedo. From these 
records the entire change in angular velocity appears 
to occur in about one or two msec, as illustrated in 
Figure 22. (This figure is obtained from records like 



this nose fitted on the CTT Steel Dummy body. These 
were 

Ui (20°) = 253 slug-ft degrees , 
a 2 = 35.5 slug-ft . 

It is clear that for this 90° cone angle on a hemi- 
sphere, with the flow separating at roughly the same 
position as on the Mark 13 nose, the whip at zero 
pitch is greater than for a hemisphere, and, as ex- 
pected, the whip changes more rapidly with pitch 
































PI 

TCH 

ANGL 

.E-T 

ME ( 

:uRv 

'F . 









!>> 



















0 4 e 12 16 20 24 28 32 36 40 44 46 52 56 

TIME IN MILLISECONDS 

Figure 22. Typical velocity and pitch angle versus 
time data from flare records (CIT Steel Dummy). 


L 


66 


WATER ENTRY 


that illustrated in Figure 14.) On the basis of the 
ideas developed, this time is much too short. For 
Ye = oOO ft per sec, it has been indicated that the 
angular velocit}^ should continue to increase for 
about 8.6 msec. However, there is also observed a 
time delay of about 5 msec between impact and the 
first motion of the flare, which indicates that at 
entry the torpedo behaves like an elastic body. This 
fact may account for the apparently too short dura- 
tion of the whip. There is another indication of the 
time duration of the whip from preliminary observa- 
tions with shock-mounted accelerometers in the nose 
obtained at the CIT Torpedo Launching Range. 
From these it is found that when Vg = 300 ft per sec 
the transverse velocity for the CIT Steel Dummy is 
about 3 ft per sec in the first millisecond. For zero 
pitch at about 300 ft per sec, Aco = 70 degrees per sec. 
Since the instantaneous center of rotation is about 
9 ft from the accelerometer, the transverse velocity 
at the accelerometer at the end of the flow-forming 
stage is about 11 ft per sec, which is much larger than 
is obtained in 1 millisecond. In fact, the accelerometer 
records indicate that in the time corresponding to 
the duration of the flow-forming stage, the transverse 
velocity will be in agreement with the value pre- 
dicted by the whip. 

B}" means of an optical whip recorder and the one- 
inch model of the CIT Steel Dummy, the Alorris 
Dam Group has obtained very fine time resolution of 
the whip during the flow-forming stage. Typical 
curves of pitch angle versus distance from entry were 
presented in Figures 16, 17, and 18. On these figures 
the distance of travel from entry, corresponding to 
the flow separating from the top of the nose, has been 
indicated. Clearly, this distance depends on de so that 
it differs in Figures 16, 17, and 18. In fact, it varies 
like cot de. From these figures it is seen that the slope 
of the curve, which is a measure of the whip, con- 
tinues to increase, although not uniformly, up to the 
end of the flow-forming stage. This is especially clear 
in Figure 16 where de = 12° so that the torpedo has to 
travel a fair distance for the flow to separate from the 
top of the nose. Since these results have been on one- 
inch models where discrepancies appear between the 
whip observed and the full-scale results, they are 
certainly not conclusive. However, they do lend 
strong support to the statement that the whip will 
increase (due to the lift on the nose) during the entire 
flow-forming stage. 

This indicates that the forces producing the whip 
are as described earlier, namely, h 3 "drodynamic (T^-) 


forces acting on the lower part of the nose while the 
upper part is still in air. Further evidence along these 
lines arises from the fact that, although the drag ring 
will markedH decrease the elastic wave pressure, it is 
seen from flare records at the CIT-TLR that the 
drag ring does not alter the whip at entiy. Conse- 
quenth', it is believed that the V- forces, which are 
probabh" unaltered b\" the drag ring, produce the 
whip at entiy. 

These observations also indicate the care that must 
be taken in interpreting observations. The torpedo is 
an elastic rather than a rigid boch", and due account 
must be taken of this fact. 

Since in the vicinit}" of the point of impact the 
shape of the cone nose tested at the CIT-TLR is 
close to a hemisphere and since the whip is still 
markedH different from that of a hemisphere, we 
have further evidence that the F- t^^pe of forces pro- 
duce the whip rather than elastic wave forces. Thus 
the elastic wave forces exist in the neighborhood of 
the impact point and should roughh^ be the same for 
both noses. Consequenthq if the elastic wave pres- 
sures produced the whip, its magnitude and depen- 
dence on pitch would be approximate!}^ the same for 
the cone and the hemisphere, while, if the forces 
produced the whip, it will differ markedl}" for the two 
heads, as is seen to be the case. 

Design Features Ixfluexcixg Whip 

It should be remembered that for all heads the 
whip is proportional to Ye. In addition, for all tor- 
pedoes with a given nose the whip varies inversely as 
the moment of inertia of the torpedo. Consequent^, 
torpedoes with greater moments of inertia will possess 
smaller whips at entiy, and torpedoes with smaller 
moments of inertia will have larger whips at entry. 

As has been seen, the nose forces producing the 
whip at entiy will vaiy considerabh" with the nose 
shape. However, for a given nose shape some indi- 
cation can be given of the variation of these forces 
with the dimensions of the nose. For example, in 
general the whip will differ for a hemispherical nose 
of large radius and of small radius. The indication 
b’om previous discussions is that usually the nose 
forces producing the whip at entiy for a given nose 
shape will increase in magnitude with larger nose 
dimensions. The reason for this is that the forces 
producing the whip, and hence the whip, appear pro- 
portional to the area of the nose. In addition, the 
whip appears to be caused hy unbalanced forces 
acting on the nose of the torpedo. The time these 


FLOW-FORMING STAGE 


67 


hydrodynamic forces remain imbalanced is the time 
for separation of the flow from the nose, which is the 
time duration of the flow-forming stage. This has 
been calculated in Section 6.2.1 and from that section 
it is seen that the time increases with the dimensions 
of the head. Consequently, we can say that for a 
giyen nose shape the impulse and forces producing 
the whip at entry will increase with increasing head 
dimensions, roughly as the third power of the linear 
dimensions of the head. In all cases the time of the 
flow-forming stage, and hence the time of the whip, 
decreases with increasing trajectory angle at entry. 

The nose shape and size determines the forces pro- 
ducing the whip at* entry; howeyer, the characteris- 
tics of the torpedo also determine the magnitude of 
the whip at entry. 

As was pointed out, for a giyen set of entry condi- 
tions increasing the moment of inertia I will decrease 
the whip at entry. Thus, eyerything else remaining 
the same, the whip at entry is inyersely proportional 
to the moment of inertia. 

In addition, it is clear that the moment arm about 
the center of grayity of the nose-lift and nose-drag 
forces will be proportional to the distance from the 
center of grayity to the nose of the torpedo h. Hence, 
for a torpedo with giyen head shape and size and fixed 
/, the quantities Ui, a 2 , as, and hence the whip, will 
be proportional to h. 

We may say that, for a torpedo with a giyen den- 
sity and fixed shape (primarily a fixed length diam- 
eter) since the moment of inertia is proportional to 
the fifth power of the length, and since the density 
remains fairly constant, the whip will yary inyersely 
as the first power of the linear dimensions. 

6.2.6 Ricochet 

In the discussion of the flow-forming stage of the 
motion it has been tacitly assumed that the head of 
the torpedo will be submerged sufficiently for the 
flow to separate and the torpedo head will continue 
to submerge. Actually, this is not necessarily true. 
Thus, when part of the nose becomes submerged, it is 
possible for the hydrodynamic lift forces to be suffi- 
ciently large so that the torpedo “bounces” off the 
water or ricochets. Ricochet means a motion such 
that the torpedo is neyer completely coyered by 
water. Of course, it is expected that a ricochet will 
occur only for yery shallow trajectory angles at 
entry. 

Obseryations of the ricochet of spheres indicate 


that the trajectory angle at entry, which must not be 
exceeded if ricochet is to occur, is giyen roughly by 
18° \/ ps, where ps = densit}^ of the sphere. As a mat- 
ter of fact, this is the underlying principle of a Ger- 
man weapon “Kurt” (a sphere which enters with de 
yery small and at high speed so that it “bounces” 
along the water surface). The sphere would land on 
the water with de = 6° and continue to ricochet until 
it lost sufficient speed. 

Concerning the ricochet of a hemispherical-nose 
torpedo, the British could not make their 18-inch 
torpedo ricochet for de down to 10° and Ve = 250 
knots. In order for the Mark 13 torpedo to ricochet, 
it is seen from the rough relation 18°/ \/ ps that yery 
approximately de must be less than 4.4 degrees. Thus, 
for yery shallow entry angles, it is possible that a tor- 
pedo with a hemispherical head will not enter the 
water but will ricochet. 

For torpedoes with ogiyal heads there will probabh^ 
be a greater tendency to ricochet; also, a nose-up 
pitch at entry will produce a greater lift force and 
hence increase the tendency to ricochet. A hemi- 
sphere would not be influenced by the nose-up pitch 
as would an ogiyal head. In general, for ricochet yery 
shallow entry angles and high entry yelocities are 
required and also nose shapes that haye large nose-up 
whips at entry. Xoses that whip down at entry prob- 
ably will not ricochet at any entry angle. 

6.2,7 Essential Data and Further Research on 
This Stage of the Motion 

Because of its great influence on the future motion 
of the torpedo, the most important quantit}^ which 
should be studied during this stage of the motion is 
the whip at entry. 

Compared to the whip all other characteristics of 
this stage appear to be relatiyely unimportant. As 
we haye seen, the whip depends both on the nose 
shape and on the torpedo body. Since the dependence 
on the body parameters are known, it is necessary to 
obtain the dependence of the whip on the nose shape. 
Hence, it appears profitable to tabulate the whip for 
yarious noses in the form of a nose-lift coefficient and 
determine the dependence of this coefficient on the 
entry conditions. 

At present it appears that perhaps the most fruit- 
ful approach to a determination of the nose-lift co- 
efficient is by direct experiment with yarious nose 
shapes. In this connection the success of some method 
of modeling the entry whip appears yery important 





68 


WATER ENTRY 


as smaller models rather than full-scale tests might be 
used. The most important unsolved theoretical prob- 
lem in water entry is a theoretically sound determin- 
ation of this lift coefficient. At the present stage of 
the development of the theory, the great analytical 
difficulties that arise in a determination of the nose- 
lift coefficient make it appear that a semi-theoretical 
experimental approach will be more profitable. 

Thus future research in water entry should attempt 
to obtain nose-lift coefficients for various shaped 
noses and the dependence of the nose lift on the entry 
conditions. 

6.3 MOTION IN THE CAVITY WITH 
THE NOSE IN CONTACT 
WITH WATER 

The time duration of the stage of the motion which 
begins when the flow separates from the top of the 
nose and ends when the tail strikes the side of the 
cavity will depend on the shape and size of the cavity, 
on the orientation of the torpedo at the end of the 
flow-forming stage, on the head shape of the torpedo, 
and on various other physical characteristics. 

In this part we shall discuss first what happens to 
the water during this stage of entry and then con- 
sider the forces acting on the nose of the torpedo. The 
determination of the behavior of the water and of the 
forces on the nose requires some knowledge of the 
pressure distribution about the nose. Finally, the 
trajectory and behavior of the torpedo in this stage 
will be discussed, with some mention of the damage 
sustained. 

6.3.1 Flow Separation and Pressure 

Distribution about the 
Nose of the Torpedo 

Remarks ox Flow Separation 

As has been mentioned, the flow separates from all 
sides of the nose of the torpedo, and in this stage of 
the motion the torpedo travels in a cavity. The water 
will remain in contact with the nose from the tip 
around to where the total pressure of the water on 
the nose falls to zero because of the velocity. Ob- 
viously, if the pressure falls below zero there will be 
no force to keep the water in contact with the nose. 

In order to formulate this condition more (juanti- 
tatively, we must consider the pressure acting on the 
nose. For the quasi-stationary flow conditions which 


exist in this stage, the integral of Bernoulli’s equa- 
tion for an incompressible nonviscous fluid is 


P — PQh + Fo + -^pl “ “ ^ P^‘ ) (12) 


where P is the pressure at any point on the nose, v is 
the velocity of the fluid relative to the torpedo at 
this point, and pgh is the gravitational static head. 
Pq is the external pressure on the fluid; it is atmos- 
pheric pressure for all full-scale launchings, while for 
some model experiments in pressure tanks it may be 
controlled and differ from atmospheric pressure. V is 
the velocity of the torpedo relative to fluid at rest. 

The flow will separate from the nose if P = Pc, 
where Pc is the pressure in the cavity. In this stage of 
the motion. Pc is probably very close to the external 
pressure Po since the cavity is open to the outside. 
However, it may be somewhat less than the external 
pressure in the neighborhood of the torpedo nose 
since the air may be circulating in the cavity. 

When P = Pc, equation (12) may be written in 
the form 


I pgh + Po — Pc 


(13) 


The quantity 

_ pgh “h Po — Pc 

is defined as the cavitation parameter (also called 
cavitation number and cavitation index) . 

In the expression for K, the numerator consists of 
pressures which tend to prevent a cavity and to keep 
the flow in contact with the nose, while the denomi- 
nator may be regarded as the pressure which tends to 
open a cavity. 

Except for a possible circulation of air which may 
be due to air rushing in after the torpedo at entry. 
Pc = Po, and, when the torpedo is still at the surface, 
h = 0 so that K = 0. As the depth increases, assum- 
ing Pc = Po, K increases. In general, even when the 
cavity is open to the atmosphere K > 0, but not very 
much greater, as may be verified by inserting appro- 
priate numerical values. For example, in Figure 23 for 
the Mark 13 torpedo with de = 20° and Ve = 500 ft 
per sec, K is plotted against distance along the tra- 
jectory for the cases of Pc = Po and Pc = 0. 

Pressure Distribution about the Nose 

The flow will separate when P = Pc or when the 
excess of the hydrodynamic pressure over the exter- 


MOTION IN CAVITY WITH NOSE IN CONTACT WITH WATER 


69 


nal pressure around the nose of the torpedo is zero. 
In order to determine at what point on the nose this 
will occur, it is generally necessary to know the pres- 
sure distribution around it. A knowledge of the pres- 
sure distribution about the nose of the torpedo is 



0 10 ZO 30 40 50 60 

DISTANCE FROM ENTRY IN FEET 


Figure 23. Mark 13 torpedo cavitation parameter 
versus distance from entry (before tail slap). 

generally necessary, not only to determine where the 
flow will break away from the nose, but also to know 
the forces and moments acting on the torpedo during 
this stage of the motion. 

The problem is quite complicated for, by classical 
hydrodynamic theory, even the steady-state equa- 
tions of motion predicting a cavity lead to nonlinear, 
integro-differential equations that are exceedingly 
difficult to handle. Most attempts at obtaining the 
pressure distribution about the nose of the torpedo, 
the shape of the cavity, the drag force, etc., deal with 
a closed cavit}^ which is usually the flow about a 
larger body than the one under consideration, and 
knowingly neglect the fact that the cavity is closed 
at the rear. They then regard the pressures and the 
flow as of importance only up to the point where the 


flow breaks away. In addition, these attempts con- 
sider only steady state phenomena. Some of these 
attempts will be briefly discussed : 

Potential Flow Method. One general attempt origi- 
nated by Shaw^ has been based on the assumption 
that the pressure distribution about the nose of the 
torpedo in a cavity is the same as the pressure dis- 
tribution with potential hydrodynamic flow about 
the entire torpedo when moving in a steady state 
through the water without a cavity. The flow and 
pressure distribution around the nose of the torpedo 
are assumed to be unaltered by the cavity as long as 
the pressure is positive. Then the point where the flow 
breaks away is easily calculated for we know P in 
equation (12) and the point on the nose where 
P = Pc- By integrating the pressure over the nose of 
the torpedo up to the point where the flow breaks 
away, we obtain the drag and lift forces and mo- 
ments. 

Originally this was an ad hoc assumption. How- 
ever, now there appears some experimental justifica- 
tion of it. The pressure distribution over various 
shaped noses fitted on a cylinder has been obtained 
in a water tunnel for various values of the cavitation 
parameter /v, although it was practically impossible 
to lower K below the value of 0.20. However, since 
a series of pressure distributions are given for values 
K > 0.20, it appears possible to determine the de- 
pendence of the pressure distribution on K. Figure 24 
is an example of such a pressure distribution. It is 
clear from the results of this work that, except for the 
finer conical noses, the pressure distribution in the re- 
gion where P > 0 is practically unaltered by changing 
the value of K. In addition, from all the results it is 
clear that, assuming a linear extrapolation to /v = 0, 
the point of zero pressure on the torpedo head can be 
located and the flow will break away at that point. 
These two observations both seem to justify the 
method of using the potential flow {K ^ oo ) about 
the nose of the torpedo without a cavity to obtain the 
pressure distribution with a cavity (/v -> 0) . 

Of course, a cavity may exist for values of K con- 
siderably greater than zero. At the surface K 0, 
but as the torpedo goes deeper K increases, and the 
pressure changes correspondingly. 

As an example of an analytical use of this method 
we shall apply it to a sphere. We find from hydro- 
dynamic theory that the pressure distribution about 
a sphere in steady flow is 


P = pgh + Po + - f pV^ sin2 rj , (14) 


70 


WATER ENTRY 



Figure 24. Effect of cavitation upon the pressure distribution around a cylindrical body with hemispherical head. 


where rj is the angle between the outward normal to 
the surface at the point under consideration and the 
direction of motion. The flow will separate when 
P = Pc, or at the angle where 


sin2 r) = ^(1 -\- 


Pq p(/h — Pc 


a + K), ( 15 ) 


or at the angle 

r)s = sin-1 (f) (1 + Ky\ 

Provided the flow does not reform on some part of 
the torpedo aft of the nose, the drag is given by 




P • dA 


over 

wetted area 


-f. 


V8 


^pV'^ I 27 rr“ sin 77 cos 77 (1 + /v — ^ siiP 77) drj 


The drag coefficient is defined by 


C’d = A F* 


SO that 


Co = [ ( 1 + /v) sin- 7?s - f siid 77 ] . 


MOTION IN CAVITY WITH NOSE IN CONTACT WITH >VATER 


71 




Figure 25. Effect of cavitation upon the pressure distribution around a cylindrical body with 45° conical head. 


Therefore, from (15) 

= + ( 16 ) 

This Potential Flo^y Method permits the calcula- 
tion of pressure distribution, point of flow separa- 
tion, drag, lift, and moment on any head if the pres- 
sure distribution from potential hydrodynamic flow 
is known. The latter may be obtained by calculation 
for simple shapes, as has been illustrated for a sphere. 
However, the essence of usefulness and simplicity of 
the method is that the pressure distribution may 


always be obtained by wind or water tunnel measure- 
ments of the pressure distribution about a noncavi- 
tating projectile. 

In general, this method is applicable to all shaped 
noses at arbitrary pitch or yaw angles as long as the 
flow does not reform at some point further aft. 
Clearly, if a torpedo head is at pitch or yaw angle, 
there will be a drag and lift force acting on the head 
and a moment about the tip of the head. If one ob- 
tains the pressure distribution from a wind or water 
tunnel, a numerical integration will give the desired 
coefficients. In some cases, such as fine cone heads. 


72 


WATER ENTRY 


the distribution of positive pressure appears to 
change with K as shown in Figure 25. In such cases 
either the distribution must be extrapolated toward 
A" = 0 before integration or the hydrodynamic coeffi- 
cients extrapolated toward A = 0. The latter appears 
the easier method. 

Approximate Method for a Sphere. There is perhaps 
only one result, obtained by G. I. Taylor, in which a 
cavity is predicted by potential hydrodynamic theory 
(for a nondecelerating body). In this result the flow 
about a paraboloid, which would have an infinite 
drag, is combined with a flow about the body with zero 
drag to produce the flow about a body which ap- 
proximates a sphere with finite drag and a cavity. 
The method is essentially that of combining a series 
of infinite line sources with a steady stream. By this 
method the approximate cavity flow about a sphere 
is obtained, but has not been obtained for other 
shaped noses. 

Source in Infinite Stream. The cavity flow about a 
sphere has been approximated (in reference 9) by a 
point source in an infinite stream. Here again, the 
drag is obtained by integrating the pressures up to 
the point where the pressure falls to zero. By this 
method the pressure is given by 


P = P„ + pgh + l-pF^ 


2hV cos g — b‘‘ 


(17) 


where b = f X radius of a sphere, r is the distance from 
the point source to the point at which the pressure is 
being measured, and rj is the azimuthal angle meas- 
ured from the direction of motion with the origin at 
the point source. By a very similar method to that 
employed for potential flow one may easily calculate 
the point of separation and the drag coefficient. 

Semiempirical Method. Recently a semiempirical 
attempt at calculating the pressure distribution was 
made. In this method a form of the pressure distribu- 
tion is assumed with constants which are adjusted to 
produce agreement with experiment. In this attempt 
it is assumed that for A = 0 : 


P 


1 

2 



B sin jSi 


1 - 6sinM/3i - /3) , 


(18) 


where A = a constant independent of head shape, 

/3 = angle between direction of motion of the 
torpedo and the tangent to the torpedo 
nose at the point under consideration, 
jSi = semiangle at the nose of the torpedo 
(/Si = 90° for a sphere), 


h = constant for a particular head shape, de- 
pending on the angle at which the flow 
separates from the head. 

Probably numerous other expressions could be 
written which, with the substitution of suitable de- 
termined constants, would give adequate approxima- 
tions to the pressure distribution. In each case, how- 
ever, such expressions must be verified by comparison 
with some other method so that it is ultimately useful 
only if it is especially convenient for integration. 

A Variational Alethod. One result has been ob- 
tained for the flow about a sphere in a closed cavity. 
By means of a doublet and a series of sources, a flow 
is set up, and then, by means of a variational method, 
the cavity is shaped so that the boundary condition 
(namely, that the pressure on the free streamline is 
the same as at the point of separation) is satisfied. 
By this method the separation point is chosen ini- 
tially, and a corresponding value of A results. Thus 
far, not many results have been obtained as the 
method is quite lengthy. In addition, it has been ap- 
plied only to the case of a sphere. 

Noses with Discontinuities. Some separate con- 
sideration must be given to noses with discontinui- 
ties. By a discontinuity is meant a sudden change in 
slope (such as a cone on a cylinder) at some point of 
pressure on the head so that, if the water is to re- 
main in contact with the torpedo, it must flow around 
a corner. The determination of the point of flow 
separation for these heads does not require a knowl- 
edge of the pressure distribution. 

From hydrodynamic theory it may be proved that 
at such a discontinuity the velocity of the water 
relative to the nose of the torpedo becomes infinite 
(limited only by viscosity), and hence the pressure 
becomes negatively infinite; that is, infinite suction 
of the water is required to cause it to flow around a 
discontinuity without leaving the surface about 
which it is flowing. 

Of all these suggested methods of estimating the 
pressure distribution, the only one that appears 
really useful is the potential flow method. This is 
really a method for interpreting pressure-distribu- 
tion measurements that can be made under noncavi- 
tating conditions in terms of the pressures that exist 
in a cavity. In the last analysis its justification is 
empirical, and it is not really a theoretical method. 

Point of Flow Separation 

As has been indicated above, the point of flow 
separation is just one aspect of the pressure distribu- 


MOTION IN CAVITY WITH NOSE IN CONTACT WITH WATER 


73 


tion and can be seen when the pressure distribution 
is known. For a hemisphere the potential flow method 
gives separation at r; = 42°, and water tunnel meas- 
urements give 45°. The other methods that seem to 
have less satisfactory foundations suggest values be- 
tween 00 ° and 75°. The exact point of zero pressure 
may be varied considerably without affecting the 
integrated forces and moments appreciably. 

The point of separation for ogives and spherogives 
can be determined by water or wind tunnel measure- 



Figure 26. Flow separation on 1-in. model of the 
Steel Dummy with a conical nose smoothly joined to the 
torpedo. Notice that the nose appears to be the only 
part of the model in contact with the water. 


ments. If the spherical nose of a spherogive has an 
angle greater than 45°, the separation will probably 
occur on it and be entirely independent of the ogive 
that follows. 

In the case of cone heads the flow will break away 
at the shoulder of the cone. This is illustrated in 
Figure 26. Similarly for flat heads, kopfrings, and 
truncated cones the flow will break away at the dis- 
continuity. 

Kinematic Theory of Cavity Shape 

As long as only the nose of the torpedo is in contact 
with the water, the behavior of the rest of the cavity 
is of little importance. However, the torpedo soon 
falls over so as to be in contact with the cavity wall, 
and it is important to know something about the 
cavity size and shape to tell how far the torpedo will 


fall. In undertaking a theory based on the laws of 
hydrodynamics, very great difficulties are encoun- 
tered and no significant progress has been made. The 
best that can be done is an empirical description 
based on an observation by Blackwell® that the diam- 
eter of the cavity at any point can be described as a 
parabolic function of the time. 

The observations upon which the description is 
based were made on the cavities due to spheres 
dropped vertically into water. It was observed that 
the water started to move radially with a velocity 
proportional to the velocity with which the sphere 
passed the point in question. Hence there may be 
defined a coefficient X such that 

Initial radial velocity of the cavity 

Velocity with which the torpedo nose passed 
this point 


The second observation was that the radial velocity 
of the cavity wall decreased with a deceleration pro- 
portional to the depth. 

To apply this kind of description to a torpedo entry 
cavity it is necessary to neglect many things, some of 
which may be important. Gravity is neglected, and 
the torpedo is considered to follow a straight path at 
an angle de with the horizontal. Furthermore, the 
effect of the water surface is neglected, and the cavity 
is regarded as a surface of revolution about the axis. 
Any effect due to air pressure inside the cavity is also 
ignored. 

The axis of x is taken along the axis of the cavity 
with the origin at the surface of the water. The radius 
of the cavity y is then regarded as a function of x 
and of the time. If t{x) is the time measured from the 
time at which the separation radius of the head, Vs, 
passes the point x, the above assumptions give 

y{x, t) = I's + XF(T)^(a:) — t-{x) . (19) 

2rs 

With the 7's in the denominator, the constant quan- 
tity jjL can be expected to be independent of the scale 
of the cavity if the Fronde law holds. 

Since the torpedo moves with a deceleration pro- 
portional to the square of the velocity, 

Vix) = , 

t{x) = t — — 1) , 

kVe 




74 


WATER ENTRY 


with k = CdpA/2M. Here t is measured from the time 
of impact, but the expressions are valid only for such 
t and X that t(x) > 0. 

As a result, the radius of the cavity at the time t 
after impact and at a distance x along the path is 


y(x, 0 


rs + 


t 5- 

. kVe 



fxx sin de 
2r, 





(20) 


The two constants X and p must be experimentally 
determined. X appears to be about 0.15 for a hemi- 
spherical nose. This means that the initial cone angle 
is considerably less than would be indicated by the 
angle of flow separation on the sphere. This is asso- 
ciated with the fact that the empirical description of 
the cavity covers only its major features and not all 
details. For blunter or sharper noses it is expected 
that X will be correspondingly greater or less than 
0.15. 

There is some indication that p is about 8 ft per 
sec per sec although this is only very crudely known. 

Later on we shall want to know the cavity width 
at a given value of a: at a time such that the tail of 
the torpedo reaches the point x. Let us say we want 
y{x) at a time t when a point h feet aft of the nose 
reaches the point x under consideration. From equa- 
tion (20) it is clear that we are interested in y{x, t) for 
t = {l/kVe) - 1). 


Hence 


,(,) = ^ (e- - 1) - 


k 


2kWe^rs 


It is clear from this expression that the cavity 
width at a point on the torpedo h feet aft of the nose 
(near the tail of the torpedo) will decrease with (1) 
increasing distance along the trajectory, (2) decreas- 
ing entry velocity, (3) increasing trajectory angle at 
entry. 


determine the torpedo motion. The resultant force 
can be resolved into a drag force that acts along the 
trajectory and a lift force that acts perpendicular to 
it. The effective point of application can be specified 
by giving the moment of the force about the tip of 
the nose. These forces and this moment can be de- 
scribed in terms of coefficients Cd, Cl, and Cm since 
it is assumed that all forces are proportional to the 
velocity squared. 

Hemispherical Xose 

Most of the studies of forces have been carried out 
on spheres or hemispherical noses because of the sim- 
plicity introduced by the symmetry. Since the forces 
are pressure forces they act perpendicular to the 
surface and hence through the center of the sphere. 
As a consequence only the drag force and its co- 
efficient Cd are of importance. 

Experimental Determinations. Numerous experi- 
mental determinations oi Cd have been made for 
spheres dropped vertically into water. In such cases 
the observed deceleration will depend on the density 
of the sphere since much of the momentum that is 
destroyed will reside in the water itself. One way in 
which to formulate the results is to assume an effec- 
tive mass of water that partakes of the motion of the 
sphere. The equation of motion is then written 

(.1/ + M’) — = -C°d-A V^- . (22) 

dt 2 

If now the forces on the sphere alone are con- 
sidered, one has 


M— = -Cd-AV\ 
dt 2 


where 


Cd = 


M 


M -f M' 


C° 


D 


(23) 

(24) 


Since M is proportional to the density of the sphere 
and il/' to the density of the water, this relationship 
may be written 


6.3.2 Forces and Moments Acting on the 

Torpedo Nose 

In the previous section, various methods of esti- 
mating the pressure distribution about the nose of 
the torpedo were mentioned. Since, during this stage 
of the motion, only the nose is in contact with the 
water, a knowledge of this pressure distribution will 


C% = (l + ^^Ci>, (25) 

where u is the density of the sphere and c is a pro- 
portionality constant. 

The assumptions involved in equation (25) are 
largely gratuitous and can be justified only in case 


MOTION IN CAVITY WITH NOSE IN CONTACT WITH WATER 


75 


the observations lead to a value of C°d that is inde- 
pendent of densit\v The evidence on this point is not 
entireh" conclusive, ^"alues between C°d = 0.26 and 
0.37 appear to have been obtained. 

For a torpedo with a hemispherical nose the pro- 
portionality constant c will be very small, and the 
observed value of Cd should be close to C°d- The 
extensive set of measurements at the CIT-TLR give 
Cd = 0.28 when the area used in the expression is the 
projected area of the sphere and is not the maximum 
cross section of the torpedo. 

Theoretical Estimations. The value of the drag co- 



• 




A 

✓ 





y 

✓ 



INTEGRAT 
PRESSURE 
OISTRIBUT 
IN FIGURE 

ED 

24 

✓ ^ 

INTEG 
DISTR 
UNITE 

RATED PREi 
BUTION IN 

D KINGDOM 

5SURE 


w 


'^EXERIM 

(WATER 

ENTAL RESl 
TUNNEL) 

JLTS 

y' 


^THEORETIC 

.c 

:AL POTENl 
3= (l+K)2 

TAL FLOW 















o 

o 


Z r 


O 

u 

s 

a: 

o 


.2 


.3 


.5 


.6 


CAVITATION PARAMETER K 


Figure 27. Drag coefficient versus cavitation param- 
eter for spherical nose. 


efficient based on the Potential Flow ^Method has 
been shown to be (1 -f /v)“. This coefficient has also 
been obtained by integrating the observed pressure 
distribution. The results obtained by these two 
methods are similar but differ by a constant factor. 


accessible values of K and then extrapolating to 
K = 0. Another represents similar integrations car- 
ried out in the report by Shaw.^ The third represents 
the theoretical coefficient based on potential flow, 
and the fourth represents some observed values. 

The drop-off in the observed values as K increases 
is due to the reforming of the flow on the afterpart of 
the body. Taking this into account, the agreement 
with the theory is reasonably good. The difficulty in 



OGIVE RADIUS IN CALIBERS 

Figure 28. Nose-drag coefficients versus ogive radius 
in calibers obtained by numerically integrated potential 
flow pressure distributions in region where P lYopV'^ -f- 
A' > 0. 

associating these values with those observed for tor- 
pedoes is partly due to a lack of knowledge of the 
proper value of K to use for the torpedo launchings. 


A 

0.0 

0.1 

0.2 

0.3 


I (1 + A ')2 
0 . 22 
0.27 
0.32 
0.38 


Integrated pressure 

0.24 

0.29 

0.35 

0.41 


Figure 27 illustrates some values of the drag co- 
efficient as a function of cavitation parameter. One 
curve represents the results of numerically integrat- 
ing the pressure distributions in Figure 24 for the 


Other Shaped Noses 

Figure 28 gives some values of the drag coefficient 
Cd based on observed pressure distribution over ogive 
heads. There are no suitable experimental observa- 
tions with which to compare these results. Similarly, 
by integration of the pressure distributions the other 
coefficients can be obtained. 

It is frequently desirable to treat the forces parallel 
and perpendicular to the torpedo axis rather than 


76 


WATER ENTRY 


parallel and perpendicular to the trajectory. These 
will be designated as Cd and Cl. Some values of 
these coefficients are given in the following table for 
a 1.4 caliber ogive. 


1.4 Caliber Ogive 


K 

0 

0.05 

a = 

0° 

0.10 

0.20 

Cd' 

0.120 

0.145 

a = 

6° 

0.169 

0.226 

Cd' 

0.120 

0.141 


0.164 

0.218 

Cl' 

0.046 

0.053 


0.061 

0.079 

-Cm' 

0.038 

0.046 

a = 

15° 

0.056 

0.080 

Cd' 

0.117 

0.135 


0.156 

0.203 

Cl' 

0.113 

0.131 


0.150 

0.195 

— Cm' 

0.102 

0.124 


0.146 

0.210 


Thus it appears that for K = 0 

Cd' =0.119, 

CC'a = 0.0076 , 

— Cm' / a = 0.0065 (destabilizing) . 

Other values may easily be obtained from the table. 

For a flat nose, experiments on 1-in. models seem 
to indicate Cd = 0.77. Various other values have 
been obtained by experiment, all of them being 
within about 20 per cent of this value and scattering 
about it. 

By integration of observed pressures Cd = 0.72, 
which is in very close agreement with experimental 
results. C l has not been calculated. 

It appears that the force coefficients can be esti- 
mated with fair accuracy from the measurement of 
pressure distributions in the cavitating state in water 
tunnels or with slightly less certainty from similar 
measurements in the noncavitating state or in wind 
tunnels. It is then possible to predict the motion of a 
body from measurements that can be made on such 
models. 

6.3.3 Motion of the Torpedo during 

This Stage 

The object of the study of the forces on the nose of 
the torpedo when in the cavity and of the changes in 
size and shape of the cavity itself is to describe the 
motion of the torpedo during this stage. 

After the flow-forming stage the axis of the torpedo 
possesses a whip (angular velocity in the vertical 
plane) given by 


— ai(^e) + (l2{Q^OLe , (26) 

r e 

where ae is the pitch angle at entry. Ui and are 
experimentally determined functions that depend on 
the nose shape and area. ai and a 2 probably have 
similar dependence on the trajectory angle at entry de 
in some manner which is not completely known. 
There are indications that they may vary as cot Og. In 
the horizontal plane the angular velocity of the axis 
of the torpedo is 


A(/^) 

IT 


= Cls'^e , 


(27) 


where \p is the yaw angle at entry. Probably = a^. 
It was pointed out that at the end of the flow-forming 
stage a differs from ae and xp differs from \pe, but the 
difference is usually not large. It is less than about 
0.2° for the Mark 13 torpedo. 

From these relations it is clear that, as long as the 
center of pressure of the nose forces lies forward of 
the center of gravity of the torpedo, the more nose- 
down the pitch angle is at entry, the smaller the 
whip; that is, the torpedo tends to “stub its toe.” 
With a nose-down pitch at entry the drag force on 
the nose usually produces a moment about the center 
of gravity of the torpedo which tends to make the 
torpedo rotate tail-up or nose-down, while with a 
nose-up pitch the moment of the drag forces is in 
the opposite direction, thus tending to make the tor- 
pedo rotate tail-down and nose-up. It is therefore 
clear that for a nose-up pitch, due to the large nose- 
up whip and the drag moment, the torpedo rotates 
tail-down and the tail soon strikes the bottom wall of 
the cavity; while for a large enough nose-down pitch, 
the nose-up whip is small and the drag forces will 
produce a moment which will cause the torpedo tail 
to strike the top of the cavity wall. For some inter- 
mediate value of a this reversal occurs. One of the 
results of the subsequent calculations is the magni- 
tude of the pitch angle ac separating these two types 
of motion. 

In the horizontal plane it is clear that, for a fine 
nose, a nose-right yaw will produce a nose-right 
angular velocity and a drag moment pushing the 
tail left or nose right, thus insuring that the torpedo 
tends to rotate to the left wall of the cavity. 

The behaviors in the vertical plane and in the hori- 
zontal plane are generally quite different for noses in 



MOTION IN CAVITY WITH NOSE IN CONTACT WITH WATER 


77 


which the center of curvature of the nose is aft of the 
center of gravity. In this case the quantities Ui, ao, 
and 03 are negative with a resulting angular velocity 
in the opposite direction to the usual case, that is, we 
get a stabilizing moment for this type of a nose as 
opposed to a destabilizing moment for the finer noses. 

From Figure 29 we see the forces acting on the tor- 
pedo and may readily write down the equations of 
motion. Gravity is neglected since this force is quite 



Figure 29. Forces and moments acting on torpedo 
(nose) before tail slap. To obtain forces and moments 
multiply coefficients by {p/2)AV'^. 


negligible compared to the other forces involved. In 
addition, in so far as the moment of the lift force 
about the nose is large compared to the moment of 
the drag force at the nose, the ratio — Cm' , 2C l gives 
the fraction of a diameter of the nose by which the 
lift force lies aft of the tip of the nose. Hence we 
know /i, the distance from the point of application 
of the lift force to the center of gravity. In the follow- 
ing analysis we shall consider the drag moment about 
the axis described by the moment coefficient Cm' ^ 
The equations of motion for sin a = a, cos a = 1 are, 
along the trajectory. 


where /i is the distance from the center of pressure 
of the lift forces to the center of gravity. 

Similar equations hold in the horizontal plane: 

Cl and Cm' are linear functions of a for small 
angles, and it is to be remembered that the effective 
pitch angle is 


Then 




h 


(d 





ri f ri > 
C M — U rn 


a 


h 

V 



It was found in earlier sections that for almost all 
noses Cd' is constant, practically independent of a 
and \p. Hence C Cd in equation (28) is of second order 
in a and is certainly negligible compared with Cd • 
The equations of motion then become 


dT 

dt 



(31) 


ydl 

dt 


2M 


C'l 


a (a — 0) 

F 


- CD'd)V‘^ 


(32) 


a 


- d = 




21 


{hC'l - CJrs) 


a — — {d — 6) 

F 


-}- Co'e cos (/)> F^ . (33) 


Since d/dt = V{d/ds) where s is the distance along 
the trajectory, these equations can be combined to 
give 

a" + 2Gd' - Ba = Q, (34) 




dF 

dt 


--(CV + CVa)AI 
2 


^2 


(28) where the primes indicate differentiation with respect 
to s and 


perpendicular to the trajectory in the vertical plane, 2G = - 




0/(1+ - CJr, jh- 2 Cd' 


MV '11= -t (CV - CD'a)A T2 : (29) 

(/( 2 


rotation about CG in the vertical plane. 


I (a- e) = - CV^r.AV^ 

2 2 and 

+ - Cc'e cos M U ; (30) 

2 


24/ L 


B = ^( hCi' - r.Cm')( 1 +^Cd'Ii 


21 


2lM 




Q = ^Cd'ccos^II + — C,7i). 

21 V 2M 




78 


WATER ENTRY 


F or many purposes the roll angle 0 may be treated 
as a constant in the study of pitching and yawing and 
the solution of (34) with Q a constant can be easily 
written down. If ao and ao' are the values of a and a' 
respectively when s = 0, 


a 


1 


(F + G) 





^(F-G)s 


+ 


1 


(F-G) 




^-{F+G)s 


+ (35) 


where F = (G“ > G. The initial pitch angle, 

ao, is the pitch angle with which the torpedo strikes 
the water plus a small correction for the refraction of 
the trajectory at impact. The initial rate of change, 
ao' , is connected with the pitching angular velocity 
by ao = ao/Ve- do is essentially the angular velocity 
of the whip. 

Since F > G, and in fact for the Mark 13 torpedo 
F> >G, the first exponential in equation (35) is in- 
creasing and the second is decreasing. Hence after a 
sufficient distance the sign of a is governed by the 
sign of [ (F + G) (ao — Q/B) + ao'] • If this quantity 
is positive the torpedo will swing to the bottom of the 
cavity. If it is negative the swing will be to the top of 
the cavity. 

In the limiting case, in which [ (F + G^) (ao — Q/B) 
+ ao'] = 0, the final value of a is Q/B. The pitch 
angle for which this occurs is called the critical pitch 
angle and satisfies the relationship 


Since 


a. = — 


ao 


F + (G B 


/ cq(^) + a2{d)ae 

ao = 


/ 


(3(3) 


this reduces to 


-«.(«) + ('h f(F + G) 

ac = ^ . (.37) 

T /(F G) 

This critical pitch is of great importance because it 
represents the dividing line between those launchings 
in which the torpedo swings to the bottom of the 
cavity and makes only a shallow dive and those in 
which the torpedo swings to the top and may dive 
very deep. 

A similar analysis can be made of the yawing mo- 


tion, but because of the symmetry of the situation 
the critical yaw angle is zero. 

Motion in the Vertical Plane 

At this point we may understand the reason for the 
great emphasis placed on the value of the whip at 
entry in the last part. From equation (36) it is clear 
that the greater the whip at entry the more nose- 
down the critical pitch, and hence the greater the 
nose-up whip at entry the smaller the chance that 
the torpedo will rise to the top of the cavity and, as 
we shall see later, dive deeply. Thus it is the whip at 
entry, which is obtained from Be and a^, that deter- 
mines how large a nose-down pitch angle can be per- 
mitted for finer heads before the torpedo goes to the 
top of the cavity. It should be carefully noted that 
both in the vertical plane and in the horizontal plane 
ac as well as ao and ao' in equation (36) are all inde- 
pendent of the entry velocity TV. Thus the behavior of 
the torpedo and the trajectory of the torpedo during this 
phase of the motion is practically independent of the 
entry velocity Fg. 

For noses whose center of curvature lies aft of the 
center of gravity (such as flat noses), ai < 0 and 
02 < 0 so that a critical pitch angle exists. This is 
generally nose-up since [a 2 (^) -f /(F + G) ] is usually 
positive. Hence it appears that for a flat nose ac will 
be positive, that is, even for a small nose-up pitch, 
the torpedo will go to the top of the cavity. This is 
just the reverse of what will happen with, let us say, a 
hemispherical nose. 

The critical pitch angle ac depends on Be through 
the dependence of Oi in equation (26) on Be. As has 
already been indicated this dependence is not known. 
There are indications that Oi varies as cot Be. At any 
rate, for finer noses it is clear that the absolute mag- 
nitude of ac decreases with increasing Be. 

ac depends on the roll angle through the term Q B. 
However, this term is generally small since the meta- 
centric height e is small. The magnitude of ac for 
different values of (/> for the Mark 13 torpedo will be 
given a little later. 

For torpedoes to be used in shallow water it is de- 
sirable to have ac sufficiently nose-down so that the 
torpedo will almost always swing to the bottom of the 
cavity. This is true even though the upturning tra- 
jectory leads to a broach for the torpedo can fre- 
(piently withstand a broach but not an impact with 
the bottom. 

E(|uation (37) shows the way in which the various 
design features of the torpedo affect the critical pitch 


MOTION IN CAVITY ^VTTH NOSE IN CONTACT WITH WATER 


79 


angle. The head shape affects the coefficients ai{de) 
and a^ide) as well as the various coefficients appearing- 
in F, G, Q, and B. The effect of the moment of inertia 
is also clear. 

As illustrations of the use of these equations, let us 
consider the behavior of a ^lark 13 torpedo and of the 
CIT Steel Dummy. These two bodies have the same 


Figures 32 and 33 show the yaw angle calculated in 
a similar fashion with the assumption that there is no 
initial yawing angular velocity. 

Similarly Figures 34 and 35 give the rate of change 
of pitch with distance along the trajectoiy and Fig- 
ures 36 and 37 give the corresponding rate of change 
of yaw angle. To obtain the time rate of change, these 



Figure 30. Mark 13 torpedo (neglecting metacentric 
height). Pitch angle versus distance from end of flow- 
forming stage for various pitch angles at entry (equa- 
tion 26). 



Figure 31. Steel Dummy. Pitch angle versus dis- 
tance from end of flow-forming stage for various pitch 
angles at entry (equation 26). 


external shape but different values of / and M. For 
the Steel Dummy e = 0. 

Both of these bodies have hemispherical noses so 
that Cl = Cd = Cd = 0.28 and Cm' = 0. The con- 
stants in equation (37) are then 


Mark 13 
G = 0.0033 
F = 0.059 
B = 0.0035 


Steel Dummy 
G = 0.0029 
F = 0.065 
Q = 0 


Figures 30 and 31 show the pitch angle a as deter- 
mined from equation (35) plotted against distance 
for various values of the pitch at entry ae. In plotting 
these curves the pitching angular velocity has been 
obtained from the whip as given by equation (26). 


values must be multiplied by the corresponding ve- 
locity, V = FgC 


Position of Torpedo at Tail Slap 

As the torpedo swings to one side or the other of 
the cavity it soon comes in contact with the cavity 
wall. This is an important point in the history of the 
water entry since new forces are brought into play 
that must be taken into account. The analysis thus 
far given is valid only up until the tail slap. 

Equation (21) gave the radius of the cavity at a 
point h feet aft of the nose as 


y.{x) = r, + ^ (C- - 1) - 

^ ^ k 2kWe^s 


80 


WATER ENTRY 


with iJL approximately 8 ft per sec per sec and X = 0.15 
for a hemispherical nose. The condition that the tor- 
pedo strikes the cavity wall at a point h feet aft of 
the nose is that 

b(a2 + rY + n = yi, 
or 

/ o I I o\ Vb T'b 

(a- -h ^ , 

0 


respond to the torpedo striking the bottom or the top 
of the cavity. The significance of the critical pitch 
angle is brought out very clearly in these curves. For 
ae = —2° the Mark 13 torpedo strikes the bottom of 
the cavity at about 64 ft from impact and with a 
pitch about 7° nose-up. However, for ae = —3° the 
torpedo never strikes the bottom at all but strikes 
the top about 48 ft from the impact and with a nose- 



Figure 32. Mark 13 torpedo. Yaw angle versus dis- 
tance from end of flow-forming stage for various yaw 
angles at entry. The curves are symmetrical about 
= 0 . 


where n is the radius of the torpedo at the point of 
contact. 

Figures 30 and 31 show also the value of a that 
satisfies this condition when h is taken as 9 ft and 
^ = 0. There is clearly a positive as well as a negative 
value of a that satisfied this condition. These cor- 



Figure33. Steel Dummy. Yaw angle versus distance 
from end of flow-forming stage for various yaw angles at 
entry. The curves are symmetrical about xpe = 0. 


down pitch of about 7.5°. To these distances should 
be added some 4.5 ft as the distance the torpedo 
travels during the flow-forming stage. 

In Figures 38 and 39 the distance to tail slap is 
plotted against ae for the Mark 13 torpedo for (/>« = 90° 
so that the metacentric height is zero, and for the 
Steel Dummy. The solid curve is for xpe = 0 while 


MOTION IN CAVITY WITH NOSE IN CONTACT WITH WATER 


81 


the dotted curve is for \pe = 3°. As is expected, S is 
diminished with increasing ype. On the curve of S 
versus ae for the Steel Dummy are included points of 
observed distance to tail slap versus ae obtained at 
the CIT-TLR. These points are obtained from an 
indication on sound records when the torpedo tail 
strikes the cavitv wall. Generallv, at the CIT-TLR 
xpe ~ 1° SO that these points are expected to lie near 
the curve for \pe = 0 and possibly slightly below it. 


and the angle the direction of motion of the torpedo 
at tail slap makes with the vertical plane is given by 


^2 


tan~^ 





where (*S) denotes the value of the quantity at tail 
slap. It is clear that 8 i and 82 are small when \pe 0 
and are large when ae ac. 




Figure 34. Mark 13 torpedo (neglecting metacentric 
height). Space pitching angular velocity versus dis- 
tance from end of flow-forming stage for various pitch 
angles at entry. 


Figure 35. Steel Dummy. Space pitching angular 
velocity versus distance from end of flow-forming stage 
for various pitch angles at entry. 


The agreement between theory and experiment is 
quite clear and is gratifying. Besides serving as a 
verification of the theory of the torpedo motion dur- 
ing this stage, it also offers some corroboration of the 
kinematic theory of cavity shape and the constants 
X and fjL which were used. 

The angle the plane which includes the trajectory 
and the point of tail slap makes with the vertical 
plane is given by 


For the critical pitch angle at entry we find, by 
inserting the numerical values in the expression (37), 
the following magnitudes for = 20°: 

Mark 13 torpedo 

ac = —2.67° for zero roll, 
a, = -2.31° for 90° roll, 
ac = -1.94° for 180° roll. 



CIT Steel Dummy torpedo 
ac = -2.47°. 



5i = tan ^ 


82 


WATER ENTRY 



DISTANCE FROM END OF FLOW-FORMING STAGE, S, IN FEET 

Figure 36. Mark 13 torpedo. Space yawing angular 
velocity versus distance from end of flow-forming stage 
for various yaw angles at entry. Figure is symmetrical 
about = 0. 



Figure 38. Mark 13 torpedo (neglecting metacentric 
height). Distance from entry to tail slap versus pitch 
angle at entry. 



DISTANCE FROM END OF FLOW- FORMING STAGE, S, IN FEET 

Figure 37. Steel Dummy. Space yawing angular ve- 
locity versus distance from end of flow-forming stage for 
various yaw angles at entry. Figure is symmetrical 
about \pe = 0. 



Figure 39. Steel Dummy. Distance from entry to 
tail slap versus pitch angle at entry. Points are obser- 
vations of this distance from sound records at CIT- 
TLH. 


MOTION IN CAVITY WITH NOSE IN CONTACT WITH WATER 


83 


These results appear to be in agreement with ob- 
servations at the CIT-TLR. The question of the de- 
pendence of ac on de is quite important and awaits 
the knowledge of the dependence of entry whip (or 



Oe IN DEGREES 

Figure 40. Mark 13 torpedo (neglecting metacentric 
height). Pos.sible relation between critical pitch angle ac 
and trajectory angle at entry de. It i.s essentially as- 
sumed that ac = ( -fli cot de)/[I{F -b (?) -b ao(20°)]. 



Og IN DEGREES 

Figure 41. Steel Dummy. Possible relation between 
critical pitch angle ac and trajectory angle at entry 
de. It is e.s.sentially assumed that ac = (— ai cot de)/ 
[I{F +G) -ba2(20°)]. 


based on the fact that a marked sensitivity of whip to 
pitch should be observed for de = 90°. Hence, as a 
crude approximation, if the indications are correct 
that ai oc cot de, o^c ^vill vary with de in a manner illus- 
trated in Figure 40 for the Mark 13 torpedo neglect- 
ing the metacentric height (0 = 90°) and as in Figure 
41 for the Steel Dummy. 

At the CIT-TLR a recorder is sometimes inserted 
in the torpedo which records the angle the axis of the 


TIME FROM ENTRY IN SECONDS 


z 

o 

N 

q: 

o ^ 

^ I 

X o 
t ° 
^ o 
< 
UJ 
X I 
< 

O UJ 
O UJ 
UJ or 
O. o 
QC UJ 
O O 
I- 


o 

UJ 

-I 

o 

< 


0 

2 





















\ 










\ 











\ 




























































AN6 

PITC 

ENT 

LE C 

H A 

RY V 

IF AX 

N6LE 

ELOC 

IS AT ENTRY = 25.8* DOWN 

AT ENTRY = 4.8* NOSE 

DOWN 

TY = 323 FPS 












Figure 42. Illustration of torpedo, entering with 
large nose-down pitch angle, going to top of cavity. 
Angle of torpedo axis with horizontal versus time from 
entry for dummy. 


torpedo makes with the horizontal. This instrument, 
known as the gyroscopic orientation recorder [GOR], 
will be mentioned further a little later. One record 
was obtained showing clearly a Steel Dummy going 
to the top of the cavity for the very steep pitch angle 
of ac = —4.8°. This record is illustrated in Figure 42. 
The axis orientation angle increases steadily from its 
entry value of 25.8 in distinction to the case of a tor- 
pedo going to the bottom of the cavity for which this 
angle decreases. ^ 


ao') on de. It was pointed out in Section 6.2 that there 
are indications that ai <x cot de and a 2 cc constant -f- cot 
de. In the quantity a 2 , the coefficient of cot de is prob- 
ably small, and we might expect that the constant 
term is very much larger than the cot de term. This is 


Dependence on Entry V elocity 

It should be emphasized that the motion of the tor- 
pedo expressed in terms of the distance along the tra- 
jectory is independent of the entry velocity. The 
critical pitch angle, the point of tail slap, and the atti- 


84 


WATER ENTRY 


tude of the torpedo at various points does not depend 
on whether it enters at 200 ft per sec or 600 ft per sec. 
This assumes, of course, that it is moving fast enough 
that an open cavity exists and is due to the assump- 
tion of forces proportional to the (velocity)^ 

The available evidence seems to bear out this con- 
clusion. 

Torpedo Damage 

Although it is not difficult to build a torpedo shell 
strong enough to withstand the drag force on the 
torpedo, the forces brought in to play at tail slap may 
crush the afterbody and fins. These forces increase 
with increasing angular velocity, and, as is shown in 
Figures 34 and 35, the angular velocity increases with 
increasing departures from the critical pitch angle 
and with the velocity. Hence the importance of a 
clean entry for preventing damage. 

Even though the shell may not be damaged by the 
drag forces, the sustained acceleration due to them 
may amount to a hundred times the acceleration of 
gravity and presents a vital problem in the design of 
the internal torpedo mechanism. 

6.3.4 Further Research 


the torpedo continues to slow down in a curved path 
until it is moving in a noncavitating state. 

We shall first consider the behavior of the cavity 
and its closure by the method outlined in Section 
6.3.1. We shall then investigate the equations of mo- 
tion of the torpedo during that stage and examine 
what is known about the forces necessary to describe 
the behavior of the cavity and the motion of the tor- 
pedo. Subsequently, the dissolution of the cavity will 
be treated. 


6.4.1 Cavity Shape and Closure 

Deep Closure 

In Section 6.3.1 a semiempirical approach to the 
cavity shape was developed. From equation (20) 


y{x ^ S,t) = fs + Wee 


t - — — 1) 

kVe 


fjLX sin de 
2r, 


t - — - 1) 

kVe 


(38) 


for 


t > — (e’‘‘ - 1), 
~ kV. 


Further research and experiment on this stage of 
the motion should probably deal with (1) Cavity 
Shape and (2) Nose Forces. 

In the direction of cavity shape a theoretical 
hydrod^mamic theory is most lacking. However, 
since very great difficulties appear, perhaps more 
work on the kinematic theory is warranted. This im- 
plies a tabulation of the quantity X as obtained by 
experiment for various nose shapes and further 
measurements to determine the quantity y. 

In order to obtain a better knowledge of the nose 
forces, the drag, lift, and moment coefficients for 
various shaped noses must be determined. In this 
direction further experiment is highly desirable. Also, 
some general semi-analytical method should be de- 
veloped and checked. The Potential Flow Method 
seems, at present, most likely to be useful, but its 
results must be checked. 

6.4 MOTION OF THE TORPEDO IN 
A CAVITY AND WHILE 
CAVITATING 

When the tail of the torpedo strikes the wall of the 
cavity, additional forces come into play in determin- 
ing the motion. Under the influence of these forces 


where k = Cd (pA/2M). The value of Cd depends on 
the nose shape of the torpedo. During the present stage 
of the motion, since both the tail and the nose are in 
contact with the water, obviously Cd will be larger 
than the value for the nose alone, and k will be cor- 
respondingly larger. 

One might expect that the proper way to treat the 
torpedo motion in the cavity would be to handle sep- 
arately the situations before and after tail slap. In 
the present state of knowledge, this results in a fairly 
complicated procedure that is of doubtful value in 
view of the uncertainties in the constants that must 
be used. It seems more practical to use an estimated 
mean value of k that may be called k and to treat the 
motion before and after tail slap as a unit. 

The type of mean k must be adjusted to the ob- 
served facts of the situation. For high speed rockets 
most of the underwater trajectory may be after tail 
slap. For torpedoes of the Mark 13 type, possibly the 
length of the trajectory after tail slap will be twice 
that before. The average used must take into account 
these facts. It is clear that, as the torpedo travels 
along its trajectory, the cavity width (2y) for a given 
X eventually decreases until the cavity walls come 
together. We wish to determine at what time this 
occurs and where the torpedo is at this instant. This 


MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


85 


phenomenon, which we will call deep closure, can be 
seen clearly in photographs of model projectiles 
launched into water. At a given point, the diameter 
of the cavity first increases after the projectile has 
passed and then decreases until the cavity closes. 
This closure will not occur at the same time at 
various points. For this reason one desires to find the 
time and place at which the cavity walls first come 
together, thus sealing off the torpedo from the at- 
mosphere. Call this time of first closure tc and its 
position Xc. 

From equation (38) setting ^ = 0, we find 


t = — 1) 

kV 


XT + Vx-T e^e + 2 Id x sm de 
+ rs r— • (39) 

jJLX sm de 


This equation gives the time of closure at a given 
position X. To find the time at which the cavity seals 
off, it is necessary to use the position at which the 
time of closure is a minimum. Writing 


T = kVet, 

X = kx , 

^ _ 2k~Ve-\rs 
IX sin de ’ 


Ts = - 1) , 

^ _ 2 IX sin de _ 1 4Its 

~ A’XnV 


Then equation (39) becomes 


Tc and Xc are plotted against G in Figures 43 and 44 
for some values of H. In Figure 45 ln(l + Tc) is 
plotted as a function of G for some values of H since 



6 


Figure 43. Graph determining time to first closure 
as a function of the torpedo characteristics and entry 
conditions. 

the distance traversed by the torpedo from entry to 
cavity closure is given approximately by 

Sc = lr ln(l 4- Tc). 
k 


f7g-AYi 1 Vl + // Wv ) 

T = e^ -1+ ~ -h V 1 j ^ 

2X 


To minimize T set clT dX = 0, which yields 


1 -b Y 

= i + .v+ - ^ + _ — •QO 


G 


Vl + HXe'-’^ W\+HXe^^ 


The solution of these equations gives AT(C, H). Then, 
substituting in equation (40), we may obtain 74, and 
the position of the torpedo may be calculated from 
the relation 


S = 1 ln(l + f TU) = 4 ln(I + ?;) . (42) 

k k 


Thus, if we know the value of X and /.i for a given tor- 
pedo and can select a suitable mean value of k it is 
possible to estimate the distance and time to cavity 
closure. 

To facilitate the use of these results, the quantities 


As an example of the use of these figures, consider 
the Mark 13 torpedo wdth Ve = 600 ft per sec and 
de = 20°. Before tail slap Cd = 0.28, and after tail 



Figure 44. Graph determining point along the tra- 
jectory where the cavity first closes as a function of the 
torpedo characteristics and entry conditions. 


slap Cd = 0.45. As a result, k may be taken as 
0.0143. We found in Section 6.3.1 that X = 0.15 
and /i = 8 ft per sec per sec. Hence we find = 4.16 
and H = 0.06. According to Figure 43 we find 


86 


WATER ENTRY 


Tc = 3.1 and the cavity seals off from the atmosphere 
0.36 sec after entry. From Figure 44 we see that the 
cavity closes first at a distance of 72 ft from entry. 
At the time of closure the approximate position of 
the torpedo is given by Figure 45 and is seen to be 
about 101 ft from entry. An entiy velocity of 350 ft 
per sec for the ]\lark 13 torpedo makes tc = 0.40 sec, 
Xc = 51 ft, Sc = 77 ft. For the same entry conditions 



01 .2 3 .4.5.6 .8 1 2 3 4 5 6 8 10 20 30 405060 80100 

Figure 45. Graph determining position of torpedo 
at first closure. 

with the Steel Dummy, but with Ye = 600 ft per 
sec, the corresponding values are tc = 0.35 sec, 
o-c = 61 ft. Sc = 83 ft, and, with Ye = 350 ft per sec, 
tc = 0.39 sec, .Tc = 46 ft, Sc = 65 ft. The velocity of 
the torpedo at this point is given b}" 

From Figures 43 and 44 for tc and Xc it is found 
that tc decreases slowly with increasing T^c- It is also 
found that Sc (the position of the torpedo at cavity 
closure) will increase with increasing Ye. 

The values for tc agree with the sound records of 
cavity collapse as observed in rocket trials in the 
United Kingdom and with the time that the depth 
and roll recorder begins to record in the British 
18-inch torpedo. The values predicted also agree 
with values obtained in many model experiments of 
the vertical entry of spheres into water. 

Using the value of k presented above, another 
interesting and useful graph is the angle between the 
trajector\" and the torpedo when the torpedo is just 
touching the cavity wall during the open cavit^^ stage. 
This is given by (y — rb)/h and is drawn in Figure 
46. This figure is only approximate since among other 
assumptions is that of a straight line trajectory. 


Surface Closure 

Numerous times in model water-entrv studies, a 
phenomenon of what may be called surface closure 
has been observed rather than deep closure that has 
been discussed up to now. By surface closure is 
meant the phenomenon whereby the cavity closes 



Figure 46. Approximate curve of mean angle be- 
tween the trajectory and the Mark 13 torpedo when it 
is just touching cavity wall. 

near the surface due to the splash folding over the 
cavity opening forming a roof. This often occurs 
quite soon after entry compared to the time for deep 
closure. The primary cause is possibly the aero- 
dynamic force acting on the splash near the surface 
of the water. 

The time at which surface closure occurs decreases 
with increasing Yej and it appears that, as Ye in- 
creases, surface closure will occur before deep closure, 
at least for small models. It is, therefore, desirable 
to know how surface closure will affect deep closure. 
On this point there is not much experimental evi- 
dence. It is expected that after surface seal the pres- 
sure in the cavity, at least at first, will tend to fall 
below Po (external pressure) since the volume occu- 
pied b}" the entrapped air increases. In the empirical 
theory of the cavity shape which was used to deter- 
mine the time to deep closure, one of the assump- 
tions was that the pressure in the cavity is Po. How- 
ever, if surface closure occurs and Pc < Po, there is 
another force accelerating the cavity walls inward, 
and the time to closure will be somewhat diminished. 
From the water entry of torpedo models there are 


MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


87 


indications that Pc ^ Pq. In addition, deep closure 
has been observed to occur after a surface closure has 
occurred in the vertical entry of spheres in water and 
appears to be relatively unaffected by the surface 
closure. 

Just how surface closure scales up has as yet not 
been determined. However, it appears that, while it 
may occur in model tests, it probably does not occur 
in the full-scale tests; furthermore, even if it does 
occur, it appears that it probably does not affect the 
deep closure and the subsequent torpedo motion. 

6 . 4.2 Motion with Tail in Contact with 

Cavitv Wall 

When the torpedo strikes either the top or bottom 
of the cavity or possibly one side, forces at the tail 
are produced in addition to the forces on the nose 
which were discussed in Section 6.3. The tail forces 
are primarily a lift perpendicular to the axis of the 
torpedo near the tail, at some distance U aft of the 
center of gravity, and also a small tail drag. Although 
the equations of motion are essentially of the same 
form as given in Section 6.3, they are quite difficult 
to handle. During this stage of the motion gravity 
may or may not be neglected, depending on the Vg 
and the distance from entry to tail slap (essentially 
the velocity). To obtain equations amenable to sim- 
ple treatment, we shall make the simplification that 
the torpedo is resting at some equilibrium pitch and 
yaw angle against the cavit}" wall. These equilibrium 
angles are such that the sum of the moments about 
the center of gravity is zero. When the torpedo first 
strikes the cavity wall it will tend to go past this 
equilibrium position and then execute damped oscil- 
lations about it. In addition, at the tail of the torpedo 
the cavity is contracting, and also the torpedo is gen- 
erally rolling in the cavity during this stage of the 
motion. 

Motion along the Trajectory 

For the motion along the trajectory, there is the 
usual equation 


where Con represents the drag due to the torpedo 
nose, and C Dt represents the drag due to the tail. 

Equation (43) gives directly the velocity T" and the 
time t in terms of the distance s that the torpedo has 
traveled along its trajectory from the position at 
tail slap. 

r = Foe ^ (44) 


To is the velocity at the beginning of this stage of the 
motion. This is given in terms of the entry velocity 
and the distance to tail slap S by 


T^o = Vpe 


-Co. ^ S 


Also 


t = 


24/ 


CupAVo 




+ ^5 , S S , 


where ts is the time to the tail slap. 

^Motion in the Vertical Plane 

In case the torpedo strikes the cavity top or the 
bottom, it will become subject to additional forces 
in the vertical plane and perpendicular to the tra- 
jectory. Subject to these forces in the vertical plane, 
the trajectory will curve up or down depending on 
whether the torpedo is in contact with the cavity top 
or bottom. 

The equation of motion is 

= -Cl — V' + Mq 
ds 2 

= - (Cl' - Cz>'«) ^ T -^ + Mg . (4o) 

2 

The angle 6 is assumed close enough to zero to permit 
setting cos 6=1 and writing Mg for Mg cos 6. Since 
the effect of this term is small, such an approxima- 
tion appears acceptable. The coefficient Cl is com- 
posed of two parts. One represents the lift at the nose 
and is called Clh- The other is the lift at the tail and 
is called C Lt- 


M 


dV 

dt 




(43) 


but the drag coefficient Cd may be divided into two 
independent parts 

C D = Cdti + Cot , 


C L — c Ln A- C Lt • (46) 

The primed coefficients describe the resultant 
forces along and perpendicular to the torpedo axis, 
while the unprimed coefficients describe the com- 
ponents parallel and perpendicular to the trajectory. 
The relation between these quantities (in the usual 


88 


WATER ENTRY 


case where a is sufficiently small so that cos a = 1, 
sin a = a) is 

Cl = Cl "h CoOi , 

Cd' = Cd- Cloc , 

this being true for the coefficients of the nose, the 
tail, or the entire torpedo. 

To obtain a relationship between C Ln and C n, it is 
only necessary to make use of the assumption that 
the torpedo is resting on the side of the cavity at an 
equilibrium pitch angle d. The sum of the moments 
about the center of mass is then zero so that 


or is constant and g/V~ does not change too much. 
The negative sign implies a circle concave-upward, 
while if d is negative the circle is concave-downward. 
As the velocity decreases the last term in (50) be- 
comes more important and may eventually change an 
upturning trajectory to a downturning one. 

Equation (50) was based on the assumption of a 
constant d and this depends on a constant oio,. Since 
the shape of the cavity changes with the time, will 
not be constant but will decrease as the torpedo 
moves along. Equation (21) gives the radius of the 
cavity at a distance h back of the torpedo nose so 
that aw can be evaluated. 


CClu' = hCu' , (47) 

where h is the distance the center of pressure of the 
nose lift lies forward of the center of mass, and C is 
the distance the center of pressure of the tail lift lies 
behind the center of mass. 

Concerning the lift forces, two assumptions are 
made that will be shown later to have some experi- 
mental basis. It is assumed that the nose lift is pro- 
portional to the pitch angle a and that the tail lift 
and drag are proportional to the departure of the 
pitch angle from the value for which the tail of the 
torpedo just touches the wall of the cavity, This 
last assumption is clearly valid only when a > a^, 
but this is the case of importance. Hence let 

Clh =c In aCn' = Cu (a — aO and C Dt =Cdt{oi — au) . 

(48) 

The condition of equilibrium, equation (47) then 
gives the equilibrium pitch angle to be 


aw — tan aw — 

h 

Ts — Tb . X(e^^ — 1) s — 1)" /r-.N 

= + z — z (51) 

h kb 2{kVeybrs 

with yb = radius of cavity at distance b back of the 
nose. 

Vs = radius of torpedo at the point of flow sepa- 
ration from the nose, 

n = radius of torpedo at a distance b back of 
the nose, 

k = mean deceleration coefficient, 
s = distance along trajectory. 

This expression for aw can be used in equation (49) 
to get a for use in equation (50). The value of 1/R 
as a function of s can then be obtained and inte- 
grated with respect to s to obtain the angle through 
which the torpedo turns in a given length of trajec- 
tory. 


a 


CuhoLw 


Cu'h - Cin'h 


(49) 


Equation (45) then becomes 


(W^l 
ds R 





+ — ■ (50) 

Y2 


This result shows that under the assumptions made 
the torpedo moves in a vertical circle as long as d, 

Strictly, the nose lift is proportional to the effective pitch 
angle at the nose, a — li/V{a — 6) and the tail lift to 
a + lo/Via — 6). However, due to the uncertainty in the 
coefficients and for simplicity the effect of the angular velocity 
terms are omitted in this analysis. No analytical difficulties 
are present if one wishes to include them. 


Motion in the Horizontal Plane 

If the torpedo enters the water near the critical 
pitch angle ac, it may swing to one side or the other of 
the cavity so that the force due to contact with the 
cavity will be horizontal. To the extent to which 
gravity can be neglected, the motion will then be 
confined to the plane defined by the direction of this 
force and the initial direction of the trajectory. Since 
the torpedo is probably symmetrical about its longi- 
tudinal axis, the force coefficients for the horizontal 
forces will not differ significantly from those for the 
vertical forces, and the radius of curvature will be 
given by equation (50). The force of gravity will not 
affect the radius of curvature but will tend to distort 
the motion out of the plane. 


>rOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


89 


General Case 

In case the torpedo strikes the wall of the cavity 
at an arbitrary angle, the forces called into play will 
lie in the plane defined by the axis of the torpedo and 
the direction of motion at the time of impact. In so 
far as gravity' can be neglected, the motion will con- 
tinue in this plane with a radius of curvature indi- 
cated by equation (50). In this approximation, the 
horizontal displacement and the change in depth can 
be estimated for any given condition at tail slap. It is 
assumed, of course, that the torpedo does not roll 
around the cavitjx For no rolling of the torpedo, the 
radius of curvature of the path in the vertical plane 
increases slowing with increasing s. In other words, 
if the torpedo is on the bottom side of the cavity, the 
torpedo trajectory is curved (and essentially circular) 
concave-upward, with the radius of the curve in- 
creasing as the torpedo travels along its trajectory. 
Thus it becomes less concave-upward until finally, if 
the torpedo is traveling sufficiently slowly, it may 
reverse sign and become concave-downward. How- 
ever, if the torpedo is on the top side of the cavity, 
the trajectoiy is concave-downward with the torpedo 
tending to dive and the radius of curvature tending to 
increase slowly. Due to the gravitational term the 
radius of curvature tends to decrease slowly, and the 
net result depends on the hydrodynamic constants 
and entry conditions. This general type of behavior 
is due to the fact that the cavity is contracting near 
the torpedo tail and its shape is changing slowly so 
that the equilibrium pitch angle is decreasing and 
thus, also, the lift force increasing the radius of cur- 
vature. 

For launchings, like those of rockets and high speed 
aircraft torpedoes, 1% is very large, and the gravita- 
tional term may be neglected. In this case, the tra- 
jectory does not differ much from a circular arc since 
the cavity contracts very slowly, and thus the radius 
of curvature of the path increases very slowly. On 
the other hand, for sufficiently low Ve, even though 
the torpedo strikes the bottom wall of the cavity at 
some distance along the trajectory, the path may 
become concave-downward, and the torpedo will 
dive. This has been clearly observed in model experi- 
ments. 

In the horizontal plane the behavior is very simi- 
lar. For no roll, if the torpedo tail strikes the right 
side of the cavity the path is concave to the left with 
slowly increasing radius of curvature, and if the tor- 
pedo tail strikes the left side of the cavity the torpedo 


path will curve to the right with slowly decreasing 
curvature. 

The change in the trajectory angle in the vertical 
and the horizontal planes during this stage of the mo- 
tion follows immediately from the discussion of the 
radius of curvature. Thus for the torpedo on the 
bottom side of the cavity the trajectory angle de- 
creases at a slowly diminishing rate, while if the tor- 
pedo is on the top of the cavity it increases at a rate 
which may diminish or increase depending on the 
hydrodynamic constants of the torpedo during this 
stage of the motion and the entry conditions. Thus 
for torpedoes on the bottom side of the cavity, the 
pitch angle at entry being more nose-up than the 
critical pitch angle {a < ac) (unless Ve is very low), 
6 will decrease with a resulting upturning trajectory, 
while if at entry ae < oic the torpedo will strike the 
top of the cavity the trajectory angle will increase 
with a resulting down-turning trajectory. 

It is evident from the preceding discussion of the 
equation of motion solution that the motion in the 
horizontal plane depends also on the entry condi- 
tions in the vertical plane, and a similar statement 
may be made for the motion in the vertical plane. 

Illustration of Motion 

The general behavior of a torpedo going to the 
bottom of the cavity may be seen qualitatively from 
a series of photographs obtained from I -in. models at 
the Morris Dam Group, California Institute of Tech- 
nology. This is illustrated in Figures 47A to 47D. 
This series of photographs is taken from the trajec- 
tory of the 1-in. vented model of the CIT Steel 
Dummy. From these pictures, which probably do not 
duplicate all the aspects of the trajectory and cavity 
shape of the full-scale CIT Steel Dummy, one may 
nevertheless see the qualitative behavior of the tor- 
pedo as described earlier. Clearly the torpedo strikes 
the cavity wall, and the tail digs in until the torpedo 
settles around some equilibrium pitch angle, as illus- 
trated in Figures 47A and B. The torpedo then pur- 
sues a curved path (concave-upwards since it is on 
the bottom side of the cavity) at roughly constant 
pitch angles as seen in Figures 47B, C, D. The cavity 
then can be seen sealing off (deep closure) from the 
atmosphere. The trajectory after closure differs more 
markedly from the full-scale and will be discussed 
later. However, it should be noted that even after 
the closure, while a cavity (although closed) may 
exist, the torpedo path is curved. In addition, one 


90 


WATER ENTRY 



C D 

Figure 47. Illustrations of the underwater trajectory of the 1-in. vented model of the Steel Dummy with a finer 
nose, de = 19°. Ve = 105 ft per sec. 

A. 10 diameters from entry. Torpedo going to bottom wall of cavity. 

B. 22 diameters from entry. Tail slap has just occurred and torpedo tail is digging in. 

C. 67 diameters from entry. Torpedo riding on bottom wall of the cavity at roughly constant pitch angle and 
pursuing an upturning trajectory (roughly circular arc). 

D. Torpedo broaching after approximately circular path. Cavity behavior appears different from prototype. 


sees the diminishing trajectory angle and the nose-up 
pitch angle during, and at the end of, the open cavity 
stage of motion. 

Other Types of Motion during 
Open Cavity Stage 

The foregoing discussion has more or less implicitly 
assumed a type of head on the torpedo in which the 
center of curvature of the nose is forward of the cen- 
ter of gravity. However, if the center of curvature of 
the nose is aft of the center of gravity (as on a flat 
nose) then it was seen (Section 6.2.5) that a restoring 
moment acts on the nose tending to return the tor- 
pedo axis to zero pitch. In this case the torpedo may 


strike the cavity wall, but the corresponding pitch 
angle will not be the stable equilibrium pitch angle. 
After striking the cavity wall, due to the restoring- 
moment at the nose, the torpedo may rebound from 
the wall, oscillate past its equilibrium position to the 
opposite cavity wall, and then rebound from that 
wall. Thus a periodic oscillation in pitch may be ex- 
pected with an attendant periodic oscillation in the 
trajectory. This entire phenomenon has been ob- 
served in model experiments with flat noses. This is 
the type of behavior which has sometimes been called 
an oscillatory type of trajectory. 

Generally, the entire preceding discussion is for 
torpedoes in which the length to diameter ratio is 



MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


91 


not too small. If this ratio becomes small a compara- 
tively wide cavity is produced, and it is possible that 
a very large value of d must be attained for equi- 
librium (which may not necessarily" be a position of 
stable equilibrium) and the motion will tend to be 
that of broadsiding. 

It has also been assumed that h (the distance be- 
tween the center of pressure of the tail lift and center 
of gravity) remains constant during this stage of the 
motion. Generallv, this is satisfactorv. However, it 
clearly depends on the shape of the tail section of the 
torpedo and may easily involve d and the cavity 
angle. For the broadsiding motion mentioned above, 
1 2 will probably vary during this stage. 

Roll ix the Cavity 

Up to now, it has been assumed that the torpedo 
does not roll about the cavity during this stage of the 
motion. Actually, this assumption may not be com- 
pletely correct. In order to examine this point, the 
roll velocity of the torpedo must be investigated. 

During the stage before tail slap, it was observed 
that a rolling velocity of the torpedo about its longi- 
tudinal axis was produced and that this velocity is 
proportional to the yaw angle at entry \pe- This roll 
velocity was caused by the moment of the hydro- 
dynamic forces perpendicular to the torpedo axis, 
multiplied by the metacentric height. Thus for the 
Mark 13 torpedo entering right side up, since the 
center of gravity is below the torpedo axis, a nose- 
right yaw produces a clockwise rolling angular ve- 
locity, while for the torpedo upside down with the 
same entry conditions a counterclockwise roll ve- 
locity would be produced. For intermediate roll 
angles the hydrodynamic forces perpendicular to the 
torpedo axis in the vertical plane also produce a 
rolling moment. 

Thus, at tail slap, the rolling angular velocity of the 
torpedo is its roll velocity at entry, (j)e, plus the rolling 
velocity induced during the previous stage of motion. 

At tail slap, an additional rolling angular velocity 
is produced due primarily to three causes, the meta- 
centric height of the torpedo (arising in an entirely 
similar manner to the roll velocity induced during the 
previous stage of motion), the torpedo propellers, and 
the torpedo fins. We will discuss each of these effects 
separately. 

Effect of Metacentric Height. The roll velocity pro- 
duced by the moment of the hydrodynamic forces 
multiplied by the metacentric height has already 
been mentioned. Clearly, roughly the same type of 


forces will act on the nose of the torpedo as in the 
previous stage of the motion where the nose alone 
was in contact with the water. The magnitude of the 
transverse forces are proportional to xp in the hori- 
zontal plane and d in the vertical plane. 

Effect of Propellers. The effect of the propellers in 
producing a roll velocity depends primarily on their 
sense of rotation. At tail slap, the propellers first 
come in contact with the water, and therefore a 
hydrodynamic force is produced. By far the largest 
component of this force is that normal to the pro- 
peller blades. The force normal to the blades pro- 
duces a rolling impulse and hence a rolling velocity 
about the longitudinal axis of the torpedo. The direc- 
tion of this rolling velocity will depend on the direc- 
tion of the force normal to the propeller blades, which 
in turn depends on the direction the propellers are 
pitched. The torpedo experiences a rolling moment 
since at tail slap the propellers probably are not up 
to their normal running speed and, in any case, the 
torpedo velocity is much greater than the steady 
running value. For a given set of propellers, the force 
and hence the rolling velocity would be a maximum 
when the propellers are fixed to prevent rotation. 

It is evident that the direction of the force on a 
single propeller which is left-handed is such that a 
counterclockwise roll velocity is expected, while if 
the propeller is right-handed a clockwise roll velocity 
is expected. With counter-rotating propellers, with 
which most propeller-powered torpedoes are fitted, 
the problem is somewhat more complex. However, it 
is probably correct to assume that the direction of 
pitch of the forward propeller will determine the 
direction of the roll velocity induced at tail slap. 

From this discussion of the mechanism whereby 
propellers produce a rolling velocity, it may be in- 
ferred that the rolling impulse and hence the rolling 
velocity will increase with increasing propeller blade 
area, increasing propeller blade pitch, and certainly 
with increasing entry velocity Ug. In addition, the 
rolling velocity will probably also depend on how 
much the propellers sink into the cavity wall. As a 
result, one might expect an increasing roll velocity 
with a larger propeller diameter even though the 
blade area is constant. 

Effect of Fins. The third possible cause of torpedo 
roll after tail slap is the action of the water on the 
torpedo fins. If there is any relative velocity, perpen- 
dicular to the torpedo axis, between the water 
bounding the cavity and the torpedo in contact with 
it, there will be a torque tending to roll the torpedo. 


92 


ATER ENTRY 


Very little is known as to the detailed motion of the 
water at the surface of the cavity, but it is in accord 
vith the simple description of cavity shape already 
given to assume that all of the motion is in a plane 
containing the axis of the cavitv. Hence anv relative 
motion must be a motion of the torpedo in the cavity. 

If the torpedo were initially in the center of the 
cavitv and moved directlv over to make contact with 
the surface, there would be no relative motion be- 
tween the torpedo and the water, perpendicular to 
the axis. It appears, however, that the situation is not 
so simple. In the first place, the torpedo is away from 
the axis of the cavity by amounts corresponding to 
the yaw angle and the pitch angle ae at entry. In 
the second place, the pitch angle will change much 
more rapidly due to the entry whip than will the 
yaw angle. Hence, when the torpedo strikes the 
cavity surface, it will not be travelling along a radius 
but along a chord of a section of the cavity. Hence di- 
rections of roll due to the fins are to be expected as 
follows; 


Pitch at Tail Slap 

Nose-up 

Nose-up 

Nose-do^Ti 

Nose-down 


Nose-right 

Nose-left 

Nose-right 

Nose-left 


Roll Velocity 

Clockwise 
Counterclockwise 
C ount erclock wise 
Clock\\'ise 


Experimental Verifi-cation. We ma^' now undertake 
to investigate the available experimental data on the 
roll velocity produced at entry and examine the 
agreement with the qualitative and semiquantitative 
theory of the phenomena presented above. 

Practically all of the data obtained on initial under- 
water roll in this country was observed at the CIT- 
TLR by means of a gyroscope orientation recorder. 
Roll records have been obtained in which the effect 
of the metacentric height, the propellers, and the fins 
have been studied. These tests were carried out on 
the CTT Steel Dummy which, incidentally, is gener- 
ally not fitted with propellers. The effect of the fins 
was to a large extent isolated by experimenting with 
the so-called WBB Dummy, illustrated in Figure 48. 
This dummy has the same physical characteristics as 
the CTT Steel Dummy, except, as is seen, mo.st of the 
fin area is removed. With the WBB Dummy as the 
basis, the effect of metacentric height, fixed pro- 
pellers, and fins was examined one at a time. 

Since on the normal WBB the metacentric height 
is zero, there are no propellers, and the fin area is a 
minimum, we expect the observed roll velocity (which 
is defined as the maximum roll velocity observed 


vdthin the first second after entry) to be a minimum, 
but Avith some residual roll due to the fact that the 
fin area is not completely removed. The observed roll 
velocity in this case is expected to have the same de- 



Figure 4S. CIT MBB Dummy minimum fin instal- 
lation. 


pendence on yaw at entry as that due to the fins 
alone so that, in the light of our previous discussion, 
a nose-right yaw is expected to produce a clockwise 
roll velocity. These roll velocities are not however 
expected to be nearly so large as with normal fins. 
In Figure 49 the observed roll velocity for the iMBB 

» 



Figure 49. Induced roll velocity for CIT MBB 
Dummy (zero metacentric height). 


Dummy is plotted against yaw at entry, and the re- 
lationship is seen to be as expected. At the origin, the 
slope of the curve is approximately 50 degrees per 
sec per degree yaw. 


MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


93 


The next step in the measurements of roll velocity 
was to put the center of gravity 0.53 in. below the 
axis of the dummy and obtain the dependence of ^ 
on \pe- It is expected from the previous discussion 
that for a given nose-right yaw angle the clockwise 
roll velocity will increase with increasing metacentric 
height. The dummies entered with ae > ac and right 
side up. Examining Figure 50, which represents the 


horizontal plane. In addition, the effect of minimum 
fins for ype = 0 is expected to be zero since the fin 
effect varies with \pe as discussed earlier. Hence for 
\pe = 0, we expect 0 = 0. One roll record was ob- 
tained with the ]\IBB Dummy with the center of 
gravity 0.53 in. below the axis in which pe = 0. The 
record is presented in Figure 51. It is seen that the 
roll velocity is practically zero as expected. 



NOSE-LEFT NOSE-RIGHT ^ 

YAW ANGLE AT ENTRY IN DEGREES ( Vg ) 


Figure 50. Induced roll velocity for CIT MBB 
Dummy (metacentric height = 0.53 in.). 

results of these tests, indicates that the observed roll 
velocity is altered by the metacentric height in the 
manner predicted. From this figure it is seen that for 
the ]\IBB Dummy the sensitivity of 0 to pe is in- 
creased to 150 degrees per sec per degree yaw so that, 
due to the metacentric height of 0.53 in., the roll ve- 
locity is increased approximately 100 degrees per 
sec per degree of yaw. 



Figure 51. Roll angle versus time for CIT MBB 
Dummy. 


O 

z 

o 

o 

UJ 

i/i 

(L 

UJ 

0. 

(A 

UJ 

UJ 

cr 

o 

UJ 

Q 


400 


300 


i 

o 200 
O 

o 

tL 

Ul 


100 


o 

q: 














ENTRY VELOCITY* 340 FPS 

METACENTRIC HEIGHT • 0.53 IN. 




o 

o 






) 




c 

o 

1 


-4-3-2-10 I 2 3 

— NOSE-RIGHT NOSE-LEFT ► 

YAW ANGLE AT ENTRY IN DEGREES (%) 

Figure 52. Induced roll velocity for CIT ]\IBB 
Dummy (fixed Mark 13 propellers with minimum fins). 


To test the effect of propellers, the MBB Dummy 
was fitted successively with ]\Iark 13 and iMark 14 
propellers fixed to prevent rotation. The ^lark 14 
propellers are somewhat larger than those of ^lark 
13. It should be noted that the ]\Iark 13 and ^lark 14 


i 

o 

o 

UJ 

(A 

a: 

UJ 

Q. 


400 


300 


200 


100 


O UJ 
O CO 

si 

^ o 
. o 


-100 


o 

IT 


-200 


-4 


- 3 



0 

0 


< 

o ® 

°0 



! 

o , 

o 



o 

0 oo 



0 













ENTRY VELOCITY . 340 FPS 

METACENTRIC HEIGHT ■ 0.53 IN. 













o 





-2 


-I 

NOSE-LEFT 


0 I 

NOSE-RIGHT 


YAW ANGLE AT ENTRY IN DEGREES 

Figure 53. Induced roll velocity for CIT MBB 
Dummv (fixed Mark 14 propellers with minimum fins). 


On the basis of the discussion of the causes of roll 
given earlier, it is expected that the effect of the meta- 
centric height in producing a roll velocity is zero for 
0e = 0 since in this case there is no force acting on a 
hemispherical nose normal to the torpedo axis in the 


forward propellers are left-handed. The result ob- 
tained with the ]\Iark 13 propellers is illustrated in 
Figure 52 and indicates that with left-handed pro- 
pellers at zero yaw a counterclockwise roll velocity 
is produced averaging about 125 degrees per sec. 



94 


WATER ENTRY 


Thus a constant roll velocity is superposed on the roll 
velocity caused bv the metacentric height and mini- 
mum fins. In Figure 53 a similar result is observed 
with fixed ^Nlark 14 propellers. In this case the roll 
velocity due to propellers is some 300 degrees per sec 
or about 23/2 times the roll velocity due to ]\Iark 13 
propellers. This is as expected since the area of the 
i\Iark 14 propellers is somewhat greater than that of 
the Mark 13 propellers. 

To study the effect of the fins in producing a rolling 
velocity the i\lBB Dummy was launched with nor- 

«■ V 

mal fins. The observed results are presented in Figure 
54. From this figure it is clear that the fins produce a 



Figure 54. Induced roll velocity for CIT MBB 
Dumnw (no propellers and normal fins). 


rolling velocity in a manner predicted earlier. The de- 
creased influence of the yaw angle on the roll velocity 
for larger yaw angles is quite apparent. Near ype = ^ 
it is found that the slope of the roll velocity curve is 
approximately 150 degrees per sec per degree of yaw 
which represents an increase of about 100 degrees per 
sec per degree of yaw over the value observed with 
minimum fins. 

We thus see that the discussion of the causes of roll 
at tail slap appears to be verified by the available 
experimental data at least for torpedo shapes similar 
to the i\Iark 13 torpedo. 

On the basis of the observed values of the roll ve- 
locity, it is possible to estimate the extent to which a 
Mark 13 torpedo might roll around the cavit}" during 
this stage of the motion and hence the extent to which 
the plane of the trajectory might be distorted. 

It appears that this stage of the motion will not 
last longer than 0.4 sec and, if the roll velocity of the 
torpedo is less than 200 degrees per sec, it will turn 
less than 80°. This will correspond to a change in 


angular position in the cavit}" that is less than 32° 
since the ratio of the torpedo radius to the cavity 
radius is about 0.4. This value of 32° based on no 
slipping at all is an upper limit, and its low value 
justifies the qualitative picture of the plane trajec- 
tory as a first approximation. 

6.4.3 Observed Values of Hydrodynamic 
Constants after Tail Slap 

In order to obtain the magnitudes of the radius of 
curvature of the path, the time and velocity distance 
relationships, as well as the cavity shape during this 
stage of the motion, the additional drag and lift forces 
due to the contact of the tail with the water must be 
known. 

^Methods of Obtaining Cd during this 
Stage of the ^Motion 

In general, we can say that Cd = Cdu + Cnt, where 
C Dn is the drag coefficient due to the torpedo nose, 
which was discussed in detail in the last part, and Cdi 
is the drag coefficient of the tail of the torpedo. The 
magnitude of Cdi may be thought to be attributable 
to the extra drag of the tail section coming in contact 
with the cavity wall or the additional drag caused 
by the increased wetted area of the torpedo in this 
stage. Hence it is expected that for an}" torpedo Cd 
will be a function of the tail shape and structure. 
Possibh" the only method for obtaining Cd is experi- 
mentally, and the simplest method for obtaining Cd 
or Cdi is probably by obtaining a velocity distance, 
velocity time, or distance time record during this 
stage of the trajectoiy. A usable drag coefficient for a 
rocket, based on full-scale and model tests, has been 
determined. This drag coefficient is generally aver- 
aged over the entire trajectory. Hence it is expected 
that the value of Cd quoted would be slightly less 
than (CDn + Cdi) since it generally includes a small 
region where Cd = Cd« (the previous stage of the 
trajectory) and possibh", for low Ve, some region 
after cavity closure. 

Another possible method for investigating the 
value of Cd during this stage of the motion makes use 
of water tunnel tests with the torpedo at a pitch 
angle d that is estimated to be roughly the value in 
the open cavity. With the cavitation number K ap- 
propriate to this stage of the trajectory (which is 
probabh" of the order of K = 0) the value of Cd, and 
also other coefficients like Cl, mav be measured. The 
increase in the value of Cd during this stage of the 


MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


95 


trajectory is expected to occur almost discontinii- 
ously since the immersion of the tail in the cavity 
wall takes place very suddenhv 

In addition, since the wetted area of the tail usually 
increases considerably with increasing d we should 
expect Cdi and hence Cd to increase with increasing d 
during the open cavity stage. Indeed, as was men- 
tioned before, we expect (« — ck:u-). The model 
tests of the forces on a long cylinder planing on water 
have shown that C Dt is apparently a linear function of 
(a — aj as was expected. 

Observed Values oy Cd 

During this stage of the motion, Cdh is approxi- 
mately the same as the value of Cd in the previous 
stage, and CdOs a function of the tail structure. 

For the CIT Steel Dummy (which is fitted with a 
shroud ring) it was found by repeated distance-time 
measurements at the CIT-TLR that, based on bodv 
diameter, during this stage of the motion Cd ~ 0.35 
approximately. Based on the actual nose-sphere 
diameter, Cd ~ 0.42 is in fair agreement with the 
values for hemispherical-nose rockets. The value of 
Cd for the full-scale ^lark 13 torpedo with shroud 
ring is probably in the same neighborhood and may 
be somewhat larger due to an additional drag force 
introduced by the propellers. From distance-time 
curves obtained with the 1-in. vented model of the 
Steel Dummy, it is estimated that during this stage 
of the motion Cd ~ 0.33 based on body diameter and 
when based on nose-sphere diameters Cd ~ 0.38. 
Hence it appears that without propellers Cdi ~ 0.14 
at the particular {d — au) holds for that torpedo. 
Adding propellers tends to diminish d and hence 
should dimini.sh that part of Cdi due to the torpedo 
tail. However, the propellers also contribute to Cd<. As 
a very crude approximation, the value of Cd for a 
flat plate of the same area may be used for Cdp- For 
the ^lark 13 propellers this appears to be using the 
area of two blades (presumably two of the four blades 
may be completely immersed). Very roughly Cdp ~ 
0.06. Due to the propellers there is also an increase in 
Cit and as a result d will diminish somewhat, thus de- 
creasing Cdi, possibh" by 0.03. Hence, by this ver}" 
rough method, for the Mark 13 torpedo Cdi = 0.17 
and Cd = 0.45. As a notation, it is here, for the first 
time, that the shroud ring has been considered since 
this is the first stage of the motion in which the tail 
structure has made contact with the water. It is 
estimated that the shroud ring drag will be small. 
Therefore, the value of Cd for the Mark 13 torpedo 
without a shroud ring is expected to be approximately 


the same. The reason for this is that the ring surface 
makes an angle of 4° with the torpedo axis. If the 
cavity were conical, then the shroud ring would also 
make roughly a 4° angle with the water when just 
touching the cavity wall. However, at the rear of the 
torpedo the cavit}" wall begins to curve inwards with 
the result that the flow past the shroud ring will be 
roughly in the direction of the ring when just touch- 
ing the cavity wall. As the torpedo tail digs into the 



CONE ANGLE IN DEGREES 

Figure 55. Drag coefficient versus cone angle for 
cone heads on a cylinder. (Obtained in the United King- 
dom.) 

cavity wall, there is probably an angle of attack of 
the flow on the shroud ring as to result in a lift. The 
total increase in drag appears however to be neg- 
ligible. 

The value of Cd measured for the ^lark 13 torpedo 
is at the equlibrium angle \/dr + f/-. The magnitude 
of this angle is not known and is not constant during 
this stage of the motion. It is estimated that the 
magnitude of this angle is in the neighborhood of 10°, 
based on the kinematic theorv of cavitv shape and 
from gyroscopic orientation recorder records at the 
CIT-TLR of the angle of the torpedo axis in the 
vertical plane. 

Results have been obtained in the L^nited King- 
dom for the Cd of models of cone head rockets. These 
results are probably averaged over the entire trajec- 
tory, and as a result the drag coefficients observed 
are probably low for this stage of the motion. The 
results indicate that C d appears practically indepen- 
dent of the length over diameter ratio for the rockets. 
Figure 55 is a plot of the results obtained of Cd versus 
cone angle. The\^ are all for h/l = 0.4. Comparing 


96 


WATER ENTRY 


these results with the Cd observed for cone noses be- 
fore tail slap, the value appears lower. This illustrates 
somewhat the inherent inaccuracies in measure- 
ments, but primarily it indicates that Cot for a cylin- 
der is almost negligible. It appears, as is expected, 
that Cd increases with the cone angle. Results ob- 
tained for Cdh for cone noses by model experiments 
and integrated pressure distributions may be com- 
pared with the results in Figure 55, and from the com- 
parison an estimate of the tail drag coefficient can be 
made. In addition, the tail drag is shown to be small 
by the fact that a smooth velocity-time relation is 
found in these model tests over the entire trajectory 
rather than the almost discontinuous curve observed 
for the INIark 13 torpedo. 

It was also found that Cnt increases somewhat as 
the center of gravity is moved back from the nose. 
This may be attributed to a greater lift required for 
equilibrium on the cavity wall and increased sub- 
mersion of the tail. 

A rocket with the Admonitor CIT head, which is a 
head looking like a large-caliber ogive, was found by 
the British to have Cnn — 0.11 and Cd = 0.16. The 
tail of the rocket is roughly cylindrical in shape. The 
same rocket with the Armo head (a flattish nose) was 
found to have a mean Cd over the entire trajectory 
oi Cd = 0.16 and to vary between 0.13 and 0.19. Va- 
rious results for the Cd of rockets (based on nose 
area) are presented in the following table. 

Table 1. Cd for rockets with various noses 

Connecting 



Radius 

region of 

Cd (based 

Radius 


in 

nose on to 

on 

of 


calibers 

cylinder 
in calibers 

nose area) 

curvatur* 


H emispherical 7ioses 


HVAR 

0.22 

20 

0.41 



0.31 

5 

0.43 



0.25 

20 

0.43 


JMA2 

Ogive 7ioses 



11.75" AR 

0.6 

20 

0.32 


2.25 "AR 

1.4 


0.33 

50 

3.5 "AR 

1.4 


0.25 

200 


1.14 

20 

0.25 

620 

5.0 "HVAR 6 

1.4 


0.25 

170 


1.0 

20 

0.25 


I. 75" AR long 
motor 

II. 75" AR 

1.5 


0.21 

550 

short motor 

1.5 


0.20 

500 

3.25 "Mark 7 

1.8 


0.31 



2.4 


0.29 



4.0 


0.16 



The 600-lb A/S bomb was found to have Cd = 0.43. 
The nose of this bomb is roughly spherical. The 250- 
lb A S bomb has a spherical capped nose of radius 
3. It was found that for this bomb Cd ~ 0.95. 

On the basis of the results presented it may be 
possible to estimate the magnitude of Cd< for various 
shaped torpedoes. Cdi appears small for rockets. Us- 
ing the Cdh for various noses given in Section 6.3, it 
appears possible to estimate Cd for this stage of the 
trajectory and to use this value in obtaining the 
cavity behavior and the motion of the torpedo. 

Lift Coefficiext 

By far the greater effect of the torpedo tail resting 
on the cavity wall is the lift that is produced near the 
tail. From the equations of motion it appears that it 
should be possible to determine the trajectory during 
this stage of the motion if the quantities Cu (vertical 
plane) and Cct (horizontal plane) are known, as C iC 
and Ccn' were derived in Section 6.3. 

From equation (50) for the first order approxima- 
tion of the trajectory during the open cavity stage, it 
is apparent that three quantities enter into the equa- 
tion, the radius of curvature R, the mean pitch angle 
a, and the lift rate coefficient of the nose CiC . The 
quantity CiC for various shaped noses was discussed 
in Section 6.3. Thus, since the approximate value of 
Cin' is known, it is necessary to know d to determine 
1 R or to know 17? in order to determine d. In order 
to determine d without knowing i?, Cu must be 
known. Actually, the quantity that is usually meas- 
ured is the radius of curvature of the path (which is 
closely circular), and sometimes from this there is 
inferred the quantity C l- 

In tests of a series of cone head rockets (cone heads 
on a cylinder) the following procedure was used. The 
value of R was found for various cone angle noses 
with different positions of the center of gravity (and 
thus different values of /i and h in equation (47). The 
values found are given in Figure 56 (Shaw and 
Naylor^) for various values of the length divided by 
diameter, position of center of gravity, and cone 
angle of the nose. It is clear from this figure that the 
smaller the cone angle the larger the value of Cl and 
hence the smaller the value of R. This is expected 
since the smaller the cone angle the larger the nose 
lift. Also the quantity Cl varies much as might be 
expected. Thus clearly Cl increases with d and from 
equation (49) it is clear that as the center of gravity 
is moved back, since hRi decreases, d will increase 
and hence Cl will increase. In addition, it is clear 
that, as I and hence length divided by diameter in- 



MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


97 


creases, the inclination of the torpedo in the cavity 
becomes smaller due to the increased length of the 
rocket and curvature of the cavity. As a result d will 
decrease with increasing 1. 

[Many underwater trajectories have been observed 
at the CIT-TLR for the full-scale [Mark 13 torpedo 
and for the Steel Dummv. However, due to the rela- 
tiveh" short path length during which this stage 
exists and due to the large radius of curvature it has 
been quite difficult to estimate R with much pre- 
cision. 

For a hemispherical nose Cu = Con == 0.28. Also, 
since Cd = Cd — C id = 0.45 — (1 + h/U^Con^ 
= 0.45 — O.G3a (/i = 5 ft, /2 = 4 ft), we may say 



0 20 40 60 80 100 

CONE ANGLE IN DEGREES 


Figure 56. Lift coefficient vensus cone angle for cone 
heads on a cylinder. (Obtained in the United Kingdom.) 

from equation (50), neglecting the dependence on 
F, that 


R 


pA 
24/ L 



a 


24/ 


0.18 4- O.G3a 


a . 


Hence we have an approximate relationship between 
] R and d. If either of the quantities is known, the 
other is determined. As has been mentioned, from 
the underwater trajectories one cannot easily meas- 
ure R on the relatively short region from tail slap to 
cavity closure. However, one may obtain the radius 
of curvature of the trajectory from records of the 
gyroscopic orientation recorder. In the approxima- 
tion indicated by the ecpiation above, d is assumed 
constant and hence the path is circular. The gyro- 
scopic orientation recorder yields records of the angle 
that the axis of the torpedo makes with the horizontal 
as a function of distance or, as illustrated in Figure 


57, time. Since the path is assumed circular, R is de- 
termined simply by R = (sc — S)/A{d — a). In this 
manner the results listed below were obtained for cold 
shots. These values are only very approximate and 
are subject to considerable error. 


R (ft) 

Weight (lb) 

600 

1,861 

500 

1,648 

500 

1,862 

450 

1,635 

450 

1,681 


However, it is remembered that the cavity con- 
tracts at the torpedo tail during the open cavity 



Figure 57. Typical record of angle of axis of torpedo 
versus time. This record is for a cold shot. 


stage. From Figure 4G it appears that the contraction 
might be taken as decreasing A(d — a) by approxi- 
mately 3° and probably more. This is a very signifi- 
cant correction and must be applied. Making this 
correction in R and adjusting the results to a common 
weight of 2,1 GO lb, by the relation expected from the 
equations, R « 4/, we find that for the Mark 13 tor- 
pedo R ~ 400 ft. Then, from the equation above, it 
is found that for the Mark 13 torpedo roughly d = 
13°. The value of d computed for the cold shots seems 
in fair agreement with the observed gyroscopic orien- 
tation recorder records. 

With the value of R, Ad can be computed. Thus 
Ad = {sc — S)/R, or dc, the trajectory angle at cav- 
ity closure, is given by 

ae > ac , dc = de — Ad , 

ae < ac , dc = de Ad . 


98 


WATER ENTRY 


Taking a typical case where Ve = 350 ft per sec 
and 600 ft per sec, it is found that Sc versus ae for the 
Mark 13 torpedo is given in Figure 58, for Sg = 20° 
and xf/g = 0°. Altering Sg with the same ag will change 
Sc through the quantity S. 

From this figure we see that a small change in ag 
from 2° nose-down to 3° nose-down changes Sc from 
15° to 27°. 



-3-2-1 0 I 2 3 


c£e IN DEGREES 

Figure 58. Mark 13 torpedo. Trajectory angle at 
cavity closure versus pitch angle at entry. 

From the expression for d given by equation (49) 
it is possible to solve for Ci/. Hence knowing d (or 
knowing R which determines d) should presumably 
determine Ci/ by the equation 


It ) 

d 

where is determined by the nose shape. Thus Ci/ 
can be calculated for the various rockets in Table 1. 
The method by which this could be carried out is 
clear. These calculations will not all be exhibited, but 
Cu will be examined for the Mark 13 torpedo with 
and without a shroud ring based on the very rough 
values of R obtained at the CIT-TLR. 

For the Mark 13 torpedo it was found that d = 10°. 
Then during this stage atv/d ^0.5, Cin'ih 'h) = 0.35, 
Cu (Mark 13 torpedo with shroud ring) = 0.70. 

From some three cold shot launchings without a 
shroud ring it is found (correcting to W = 2,160 lb) 


that R ~ 200 ft and as a result d = 20°. Conse- 
quently, Cl/ (Mark 13 torpedo without shroud ring) 
= 0.50 is then found. 

^Try similar calculations may easily be carried out 
for the horizontal plane. 

On the basis of the various values of R and Cd = 
Cd' C Ld, one may calculate Cu and it appears pos- 
sible to make a rough guess at the values for arbi- 
trarily shaped projectiles. However, this is still very 
crude. 

The possible oscillatory behavior of flat-nose tor- 
pedoes has already been mentioned. The trajectory 
resulting is essentially oscillatory with no general 
curvature. 

Effect of Propellers during this Stage 
OF THE Motion 

The effect of propellers during this stage of the 
motion is to produce a rolling velocity, to produce a 
small increase in tail drag Cot, and to produce a cross 
force at the torpedo tail. It is this cross force that is of 
major importance since it may produce large changes 
in the underwater trajectory. If a torpedo enters with 
ag > ac SO that it strikes the bottom of the cavity wall 
and if there is a sufficiently large cross force at the 
tail, it may rebound to the top wall, and the subse- 
quent trajectory would be of the downturning type 
instead of the expected upturning trajectory. 

In model tests by the British of their aircraft tor- 
pedo it has been observed that without propellers the 
torpedo rested on one side of the cavity, while with 
propellers the torpedo rebounded from the wall which 
it struck originally. Model tests of the Mark 13 tor- 
pedo in the United States made without propellers 
indicated no rebounding. In addition, tests have been 
made with a full-scale powered Mark 13 torpedo 
equipped with propellers. From these tests a clear 
sensitivity of depth of dive to pitch angle was ob- 
served. For ag > ac the depth of dive was relatively 
shallow, while for ag < ac the torpedo struck bottom 
or at least dived very deep. If rebounding occurred, 
one should have observed some deep dives and some 
shallow dives for ag > ag. Since this was not observed, 
it is inferred that the Mark 13 powered torpedo does 
not rebound. Records obtained with the gyroscopic 
orientation recorder on Mark 13 shots under power 
indicate that during the open cavity stage there is no 
marked change in the inclination of the axis of the 
torpedo in the vertical plane as would appear with 
rebounding. 

From a single gyroscopic orientation recorder rec- 




MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


99 


ord it appears that R is practically unaltered by the 
removal of propellers. However, the crudeness of the 
measurements and the fact that there is only one 
record makes the evidence extremely inconclusive. 
Further experiments were made at the CIT-TLR 
with Steel Dummies fitted with fixed iMark 14 and 
^lark 15 propellers. For these launchings (where 
ae > oic) the gyroscopic recorder clearly indicated the 
torpedo rebounding from the bottom cavit}" wall. 


is some three feet longer, d is smaller; thus a smaller 
moment holds the torpedo against the cavity wall. 
There may also be a larger cross force on the British 
propellers than on the Mark 13 propellers. 

6.4.4 Transition Region 

The behavior of the torpedo in the transition re- 
gion, that is, from its position at cavity closure to the 



z TIME FROM ENTRY IN SECONDS 

< 


Figure 59. Inclination of torpedo axis versus time 
for MBB Dummy with fixed Mark 14 propellers. Notice 
nose-up pitching velocity at entry and rebounding from 
bottom cavity wall. 


Typical records obtained are illustrated in Figures 
59 and GO. For launchings of this type the trajectory 
data indicated a very deep dive. In addition, some 
records have been obtained with fixed propellers 
showing that the dummy rebounded from the bottom 
wall and down again from the top wall. From these 
results it is inferred that the cross force on fixed pro- 
pellers is considerably greater than the cross force on 
moving propellers. This is perhaps what might be 
expected since the relative velocity and angle of 
attack on the propellers is reduced when they are re- 
volving. In any case, it is probably true that Ci/ with 
propellers is greater than without propellers. 

As far as is known, the rebounding of the British 
torpedo has not been proved by observations on a 
full-scale torpedo. However, there are elements in 
that torpedo which make it more likely to rebound 
than the Mark 13 torpedo. The nose area of the 
British torpedo is 1.5 times the area of the Mark 13 
nose, and hence the moment holding it in the cavity 
wall is greater. In addition, since the British torpedo 


(/) 

UJ 

UJ 

(T 

O 

UJ 

O 

z 


z 
o 

N 
(£ 

O 
Z 

o 
z 
< 

<5 Q 

< UJ 

o ^ 

§ ! 
cc 
o 


Q. 

z> 


80 


60 


40 


20 


^ i 

W O 

“ Q 

UJ < 

o ^ 
z 
< 


-20 


-40 




















/ 

/ 















/ 















— p 

TCH 

VEL 

OCIT 

Y = 1 

20" 

PER 

SECC 

)ND 






/ 

/ 














/ 















/ 

/ 



METACENTRIC HEIGHT = 0.53 IN 

PITCH ANGLE AT ENTRY = 0.9’ NOSE-DOWN 

/ 

/ 



/ 





























































I 2 

TIME FROM ENTRY IN SECONDS 


Figure 60. Inclination of torpedo axis versus time for 
MBB Dummy with fixed Mark 13 propellers. Notice 
nose-up pitching velocity at entry and rebounding from 
bottom cavity wall. 


point where it is in practically noncavitating motion 
is about the most obscure part of the underwater 
trajectory. 

Extent of the Transition Region 

The point where this region begins is quite definite : 
it is the location of the torpedo at cavity closure and 
is given approximately by equation (42) and Figure 
45. The point where this region terminates is some- 
what arbitrary. The method that will be used to de- 
termine it is based on the fact that, after this transi- 
tion region is traversed, it is desired to treat the tor- 
pedo as though it were in noncavitating motion. 
Hence the termination of this transition stage will be 
governed by the fact that the cavitation parameter K 
attains a value K* such that effectively the torpedo 
is in noncavitating motion. The word “effectively’' 
has been used since water tunnel tests indicate that 
an advanced stage of cavitation must be reached for 
the hydrodynamic parameters, especially Cd, to differ 
significantly from their values for noncavitating mo- 
tion. Furthermore, since the fins of the torpedo and 


100 


WATER ENTRY 


the control surfaces are very effective in controlling 
the motion when the torpedo is in a noncavitating 
state, the value of K corresponding to the end of the 
transition region is also determined by the require- 
ment that there be no effective cavitation on the fins 
or control surfaces. Having found this value of /v* we 
may determine the approximate position of the termi- 
nation of the transition stage. Thus 


X ow roughly 
and 


^ Po — Pc 


h* ~ s* sin de , 


I"* _ ^-KscQ-k{s* - Sc) 


where Sc is the position of the torpedo at cavity clos- 
sure and is given by equation (16). k" is given by 
k" = Cd" (p-d/ 23/), where Cd' is the average drag co- 
efficient in this region. Pc, the pressure in the (closed) 
cavity, is not known and this point will be discussed 
later. However, the length of this transition stage 
(s* — Sc) is relatively insensitive to changes in Pc. It 
is therefore assumed that Pc = Po- Hence we may say 
that the transition stage terminates approximately at 
the distance s which satisfies the equation 

K* ^ 17 e"*' “ = pgs* sin $, . (52) 

2 

For example, from water tunnel tests it appears 
that for the Mark 13 torpedo K* ~ 0.35. For = 
20°, Ve = 350 and 600, it was found Sc = 77 ft and 
101 ft. Using for k" the value based on Cd = 0.25 it 
is found that s corresponding to the end of the transi- 
tion region is 113 ft and 145 ft, or only 36 ft and 44 ft 
after cavity closure. 

At cavity closure some air is entrapped with the 
torpedo. It appears that, due to the fact that the 
cavity closes in on the air surrounding the torpedo 
and since this air is at a depth corresponding to a 
considerable hydrostatic pressure, the pressure of the 
entrapped air tends to rise above atmospheric pres- 
sure Po. At the same time some of the entrapped air 
is being entrained in the wake of the closed cavity 
and transported away. This phenomenon tends to 
diminish the pressure in the cavity during this stage 
of the motion. In addition, the cavitation parameter 
K is still sufficiently low so that the torpedo is cavi- 
tating, and as the torpedo moves along this tends to 
increase somewhat the volume of the cavity. By 


cavitation is meant the phenomenon whereby the 
pressure around some part of the torpedo falls to the 
value of the vapor pressure of the water (due to the 
velocity being sufficiently high) so that local vapoii- 
zation of the water immediately commences, and 
ceases only when the pressure is greater than the 
vapor pressure of water. It is clear that the pressure 
cannot fall below the vapor pressure of water. 

Thus it is clear that there are a number of effects 
tending to alter the pressure in the cavity, and it is 
difficult to say which one predominates. It is be- 
lieved that the net result is that the pressure in the 
cavity may perhaps increase a small amount after 
cavity closure. This probably does not affect the 
motion significantly. Earlier in this section, in order 
to find the distance to where the torpedo is effec- 
tively in noncavitating motion it was assumed that 
during this stage Pc = Po- This appears sufficiently 
accurate for such a calculation since the distance is 
relatively insensitive to changes in Pc- 

The mechanism by which the air or cavitation 
cavity is carried away after deep closure is still 
obscure. It may be that small bubbles are entrained 
and carried away in the wake. 

• It appears from what has been said that the volume 
of the closed cavity diminishes since the pressure 
tends to increase and the mass of air in the cavity 
tends to decrease. Hence we may expect that the 
cavity width at the torpedo tail will probably dimin- 
ish somewhat. The magnitude of this change in the 
cavity width at the torpedo tail is not known pre- 
cisely, but as a very rough approximation the expres- 
sion for the cavity width at the torpedo tail used 
previously might be used here and apply also to the 
short transition region. At the conclusion of the 
transition region is assumed to be in the neighbor- 
hood of the value at the beginning of the region. It 
appears from water tunnel tests that for K ^ 0.25 
there is a linear relation between the maximum closed 
cavity diameter and the radius of the hemispherical 
nose, as is expected from equation (51), where at a 
given cavitation parameter y is seen to be a prac- 
tically linear function of /’«. This point lends further 
support to extrapolating the cavity width equation 
to the transition region. 

A consideration of the forces on the torpedo during 
this stage of the motion indicates that any attempt 
at estimating their magnitude other than by experi- 
ment leads to almost unsurpassable difficulties. The 
expectation is that during the earlier part of this 
transition region Cd is approximately the same as in 


MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


101 


the stage with the open cavity, since Pc = Pq, and 
that the flow has not yet reformed on the afterbod^^ 
As the torpedo progresses during this stage, the cavi- 
tation parameter K increases until the cavity no 
longer envelopes the entire torpedo but begins to 
close in and the flow to reform on the afterbody. As 
soon as this occurs it is clear that a sudden reduction 
in the value of Cd will result since there will be pres- 
sures acting on the tapered afterbody of the torpedo 


Cd, it should be noted from Figure 62 that Cd does 
not increase significantly for the Mark 13 torpedo 
without propellers until K is well below the value for 
incipient cavitation. 

The difficulty with the experimental approach to 
the problem is that with most apparatus one cannot 
reduce K sufficiently to reach the value existing dur- 
ing the transition region. Thus at the beginning of 
the transition region K ~ 0, and at the end of the 




K= 0.22 



K=0.23 




K= 0.39 



K= 0.74 



K= 0.78 


Figure 61. Illustration of change of flow in water tunnel about a 2-in. model of the Mark 13 torpedo with decreasing 
cavitation parameter, K = (Po — Pc) PA.py'^i for a pitch or yaw angle of 6°. 


in the opposite direction to the drag forces. Hence it 
is expected that the resultant drag force will continue 
to decrease as the flow proceeds to reform on more of 
the torpedo afterbody. As an illustration of how the 
flow around the torpedo changes with cavitation 
parameter K and yaw or pitch angle. Figure 61 is 
inserted. These are photographs of models in a water 
tunnel. It is clear that the size of the cavity increases 
for decreasing /v. Calculating the value of Cd is 
therefore seen to be nearly impossible due to the lack 
of knowledge of the back pressures. 

For the Mark 13 torpedo it appears from both full- 
scale and water tunnel results that C d decreases dur- 
ing the transition region. Thus Cd, as defined by the 
relation Cd = — (2d//pA) {dV/dt), diminishes from 
about 0.45 to approximately 0.1 in this small transi- 
tion interval. In connection with the magnitude of 


stage K has a value /v* which will be different for 
each projectile. As will be seen /v* 0.35 for the 

Mark 13 torpedo. Thus most of this stage of the mo- 
tion is difficult to duplicate. There is some promise in 
apparatus now being constructed of ability to meas- 
ure the forces and observe the cavitation for very low 
values of /v, theoretically, almost down to K = 0. 
Some investigation of the forces acting on various 
torpedo bodies has been carried out in the variable 
pressure water tunnel at the CIT Hydrod 3 mamics 
Laboratory. The essential results obtained are given 
in Figure 62. From these figures it is seen that pre- 
cisely in the region of greatest interest (low K) the 
coefficients Cm, Cc, and Cd (for the Mark 13 torpedo 
body) vary so rapidly that extrapolation is unreliable. 
The precision of the results is not entirely known. 

The value of /v* was chosen on the basis of photo- 






102 


WATER ENTRY 


graphs of the flow about the torpedo in the water 
tunnel for various values of K. From Figure 62 we 
might expect that roughly K* = 0.35. Actually it 
would be necessarv to measure effectiveness of fins 
and control surfaces as a function of K and to ascer- 
tain from these tests what value to use for /v*. How- 






.5X76® SF 

»HEF 

OGIV 

'E 





r 

9 

n 

^ M K 15 










//^ 

''HEMISP 

1 

HERE 






0 0.2 0.4 0.6 0.8 1.0 l.l 

K 



0 0.2 0.4 0.6 0.8 1.0 l.l 

K 


mately its value in still water at the end of this stage. 
In addition, it appears that during this stage the tor- 
pedo continues to roll with a somewhat decreasing 
angular velocity due to some damping moments. 

Since the magnitude of the forces acting on the 
torpedo during the transition stage is obscure, the 
trajectory of the torpedo during the transition stage 
may be taken, in a first approximation, to be a con- 
tinuation of the trajectory in the open cavity stage 
and may be roughly described as a curve with in- 
creasing radius of curvature. 

For the value of d at the end of the transition re- 
gion one may use as a first approximation the d dur- 
ing the open cavity stage. However, there is some 
evidence pointing to a smaller value. First of all, 
from Figure 46 we see that decreases considerably 
and quite rapidly during the transition region since 
it depends on a term of the form Secondly, from 
Figure 62 which gives an indication of the approxi- 
mate static moment coefficient of the forces on the 
torpedo, there are indications that there is close to 
zero moment on the torpedo at K ^ 0.25 and d = 3°. 
Based on these considerations it can be indicated 
that toward the end of the transition region d for the 
Mark 13 torpedo is approximately 5°. 



Figure 62. Cm, CV, and Cd versus K for various nose 
shapes. Measurements on a projectile 7.18 calibers in 
length with center of gravity 42 per cent of the length 
from the nose. 

ever, with the present apparatus it is not feasible to 
run such tests. For calculations involving Cd during 
the transition stage, the mean value of Cd (and hence 
K) during this stage is used, namely, Cd = 0.27. This 
Cd is the mean between the open cavity value and the 
value at the termination of this stage. 

During this transition stage, as the torpedo makes 
increasingly better contact with the water, the buoy- 
ant force on the torpedo increases from its value of 
close to zero in the open cavity stage to approxi- 


Condition of the Torpedo When Effectively 
IN Noncavitating Motion 

At the end of the transition region the torpedo is in 
the initial state for the subsequent run. The speed is 
given by 

V = Ye . 

The pitch angle will be determined by the nose 
shape (cavit}" width) and the values of de, Fc, and ype, 
and critically by ae. If oLe is more nose-up than ac the 
pitch angle at this point will be roughly the same 
number of degrees nose-down. For example, with the 
Mark 13 torpedo it was determined that at this point 
a = 5° nose-up for ae > ac, while d = —5° for ae<ac. 

The yaw angle will also depend on these variables. 

The roll orientation clearly depends on de, and de, 
as well as \pe, and the propellers. It is probabl}^ correct 
to say that any orientation is eciually likely with the 
probability that the torpedo is a little closer to right 
side up than upside down. 

The condition at cavity closure of the torpedo for 
which ae ~ ac (so that the torpedo never really 
strikes the cavity wall) has to be considered a little 
more in detail. Thus for a torpedo entering in this 
manner we can estimate a;(s) and \p{s), and the indica- 


MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


103 


tions are that these remain fairly close to zero. The 
tail of the torpedo does not touch water until the 
cav’ty closes in about the torpedo. It must then be 
determined at what distance a/ a~ xp- = the angle 
of the cavity. Up to this point, Cd corresponds to 
Con as given in Section 6.3. Hence, up to the distance 
S where the cavity closes in around the torpedo, the 
path is straight for a hemispherical nose, and, from 
that point to the end of the transition region, it is 
curved, depending further on whether the torpedo is 
toward the top or bottom cavity wall. For the limit- 
ing case of ae = ac, 6* = Oe, a* = ae, D = s* sin de. 
Clearly, if the cavity is closing in about the torpedo, 
the distance (s* =**S) is going to be relatively small. 
From the methods presented for handling the under- 
water trajectory, it is clear that these intermediate 
cases can be calculated. 

6.4.5 Further Research and Experiment 
during the Open Cavity Stage 
and Transition Region 

Despite the fact that a fairly clear qualitative de- 
scription of the torpedo behavior during this stage of 
the motion has been presented, the quantitative re- 
sults are very uncertain. 

It appears that future research could determine 
some of the fundamental quantities considered. To be 
more precise, future study of the water entry of tor- 
pedoes should attempt to obtain with greater preci- 
sion the Cot (tail drag coefficient) value during the 
open cavity and transition stage for various tail 
structures. Similarly, the value of Cu (tail lift coeffi- 
cient) is of importance and should be tabulated for 
various types of torpedo tails. 

Presumabhq on the basis of the theory presented, 
these coefficients might well determine the under- 
water trajectory of the torpedo during this stage of 
the motion. However, the cavity shape also enters 
into these considerations, and the theory of the cavity 
shape is only approximate. Hence during this stage of 
the motion the importance of a more precise theory 
of cavity shape is clear. It is especially important in 
the transition region since the pitch angle at the end 
of the transition region could then be determined. 

If the value of the radius of curvature were known 
with some degree of precision or if a were known, then 
Ch could be determined. 

The most promising approach to practical results 
appears to be of the experimental type. Two general 
attacks appear hopeful. In the first method, observa- 


tions are made either on full-scale torpedoes with 
recording apparatus or on reliable models with 
cameras. From these observations Cd, R, and d may 
be determined and hence Cd/ and Ci/. The second ap- 
proach is to obtain sufficiently low cavitation param- 
eters in some water tunnel arrangement so that in 
this tunnel the forces and moments on the torpedo 
may be measured and the coefficients then deter- 
mined directly. 

As far as the cavity shape is concerned, the present 
kinematic theory may prove adequate. However, 
further experimental verification and research on the 
constants in the equation is necessary. Particular 
attention must be paid to the cavity shape during the 
transition region. Thus future research should obtain 
the basic constants entering into the equations of 
motion Cd / and Ci/, as well as the change in cavity 
shape during the transition region. This will deter- 
mine the motion in this stage and the condition of the 
torpedo at the conclusion of this stage. The condi- 
tion of the torpedo at the conclusion of this stage to 
a large extent determines the subsequent motion. 

Modeling 

The study of water entry by means of small-scale 
models is very attractive since underwater trajec- 
tories and many details of behavior may then be ob- 
served at a comparatively small expense. 

The conventional method used for studying water 
entry is Froude modeling. By this method the scale 
factor for length of the model Si is the square root of 
the scale factor for time St, or Si = -xCSt. As a conse- 
quence, accelerations on the prototype and the model 
are the same. This may be seen from the fact that 
accelerations have the dimensions I on the proto- 
type and SiI/(Sity = l/f on the small model. As a re- 
sult the effect of the acceleration of gravity (or the 
weight) is properly modeled. 

This type of modeling has appeared to give realistic 
results in studying the water entry of blunt nose anti- 
submarine weapons and, in fact, for most projectiles 
studied in the past. 

However, in the fall of 1944 the illusion of the uni- 
versal success of a Froude model was shattered. As 
part of a program for studying head shapes, the CIT 
Morris Dam Group studied the water entry of a finer 
shaped nose than the hemisphere fitted to the Steel 
Dummy. With this fine nose the dummy actually 
dove to the bottom with a 4° nose-up pitch rather 
than having the expected shallower trajectory. Then 
the CIT-TLR fitted this head to a full-scale dummy. 


104 


WATER ENTRY 


and the trajectory was of the rapidly upturning type 
until a pitch angle of 1.5° nose-down was reached. 
Hence an important violent discrepancy between full- 
scale and model trajectories was observed. This ap- 
pears to be the first time that Froude modeling was 
noticed to be markedly unsuccessful in modeling 
water entry. The recognition of this fact appears to 
be a major step in the study of modeling, and the 
means of correcting this defect is the major problem 
which still appears to be wanting a complete solu- 
tion. The discussion of the discrepancies, their causes 
and methods of partial solution are best considered 
under the various stages of entry. 

Flow-Forming or Whip-Producing Stage 

In the light of the theory presented in Section 6.3, 
one might expect that the reason why the model dove 
when it was expected to turn upward should prob- 
ably be due to the fact that the whip at entry was not 
properly modeled. Hence, it was decided at the 
Morris Dam Group that the whip at entry of the 
models should be studied. To this end an optical 
whip recorder was designed whereby the whip at 
entry of the models could be measured. The results 
obtained by this recorder indicated that the whip of 
the models was radically different in many respects 
from the prototype whip. 

It is remembered, from Section 6.2, that the proto- 
type exhibits a large nose-up whip which varies as 
the first power of the entry velocity, or the whip di- 
vided by the entry velocity was a constant indepen- 
dent of the entry velocity. On the models the follow- 
ing startling results were observed : 

1. At a Froude prototype velocity of 475 ft per sec 
the whip of the model of the Steel Dummy was only 
about one-third of the whip of the prototype, thus 
indicating why the trajectories were different from 
the prototype and the critical pitch results smaller. 

2. At this same velocity, with the finer nose on the 
dummy, the Avhip at entry was actually nose-down 
instead of the very large nose-up whip observed on 
the prototype. 

3. In addition, on the 1-in. model of the Steel 
Dummy the whip was observed to increase as the 
square of the entry velocity, while with the finer nose 
it increased very much more rapidly than the square 
of the entry velocity. 

These results indicated the truly bad state of 
affairs in the modeling of the water entry of finer 
noses. 


The next obvious question was what caused this 
very anomalous behavior of the 1-in. Froude model. 
Clearly, a different set of forces was controlling the 
model behavior than was controlling the prototype 
behavior. This is perhaps most clearly seen from the 
very large dependence of the whip on entry velocity. 

It was evident that a large nose-down lift was 
acting on the torpedo nose during the flow-forming 
stage, and the various possible causes of this nose- 
down lift were systematically investigated. These 
investigations are reported in reference 14. The result 
of this study indicated that the cause of the nose- 
down lift appears to be associated with the narrow 
air space which exists when the flow separates from 
the finer shaped noses. There is a pressure drop in 
this narrow air space due to the viscous flow of air 
in it. 

For the 1-in. model the Reynolds number in this 
space is estimated to be only around 5. Rough calcu- 
lations indicate that due to this viscous flow of air 
the pressure reduction on the underside of the nose is 
of the order of Y 2 atmosphere, while on the top side 
of the nose is roughly atmospheric pressure. It ap- 
pears that the downward lift force on the nose de- 
pends on the area and thickness of this laminar space 
between the solid and liquid and hence on the curva- 
ture of the nose, as is expected from the anomalous 
behavior of the finer noses. 

Generally, it appears that the magnitude of the 
down lift varies with the Reynolds number. The 
ratio of the down lift to the normal nose-up lift caused 
by the hydrodynamic F“ forces should decrease with 
increasing entry velocity and size as well as probably 
with decreasing density of air. 

These contentions are supported somewhat by 
photographs of a launching of the finer head. One 
photograph is illustrated in Figure 63. On this photo- 
graph is clearly seen the fact that the water tends to 
‘‘stick” to the underside of the nose. This is in marked 
contrast with Figure 4 in which the flow separates 
from all parts of the model nose. 

In order to equalize the pressure in the air space on 
the underside and top side of the nose, a scheme 
which has proved moderately successful is known as 
“venting.” By this method, holes are drilled in the 
torpedo model which permit free passage of air from 
the underside to the top side of the nose. As a result 
the pressure in the air on the underside of the nose 
tends to be the same as on the top side of the nose, 
which is the desired result. Many types of venting 


MOTION OF TORPEDO IN CAVITY AND WHILE CAVITATING 


105 


were tried using different shape and size holes and 
grooves. One fairly successful vented model in which 
the nose alone was vented is illustrated by the nose 
in Figure (35. (The other vents in this figure will be 
discussed subsequently.) 

With this type of venting the model whip more 
nearly duplicated the prototype whip, although the 
problem is still far from being completely solved. 



Figure 63. Water entry of unvented fine no.se 2-in. 
model of Steel Dummy. Notice water “sticking” to bot- 
tom side of nose. Compare with Figure 4. 

With venting, the model whip reached the value of 
only 70 per cent of the prototype whip. However, 
even with venting, the whip increases faster than the 
first power of the entry velocity, which is still in dis- 
agreement with the prototype results. 

The results obtained with the Steel Dummy (with 
the Mark 13 nose) are illustrated in Figure G4. From 
this figure the discrepancy with the prototype is 
noticed. 

Another troublesome point that was observed is 
that with the 2-in. model, which, since it is larger, 
should more closely duplicate the prototype whip, 
actually exhibited a smaller whip than the 1-in. 
model both when unvented and when vented. 

Thus it is seen that, although much progress has 
been made in understanding the modeling of the 
entry whip, the state of the problem is still far from 
being satisfactory and discrepancies with the proto- 
type results still remain. However, with blunter noses 
and with heads possessing discontinuities small 
models will probably not exhibit this very anomalous 
effect to any marked extent. 

From the present understanding of the cause of 


the smaller whips in 1-in. models, one might expect 
that the modeling will improve with increasing entry 
velocity, thus indicating that stress modeling {Se = St) 
is perhaps much better than Fronde modeling during 
the flow-forming stage. Decreasing the density of the 
air also seems to be a step in the right direction. 

Up to Tail Slap 

The lack of modeling of this stage is probably due 
to lack of modeling of the entry whip. 


Cavity Behavior and Trajectory after Tail Slap 

In the oblique entry of 1-in. models surface closure 
is practically always seen to occur. This may not be 


0.22 


0.20 


0.18 


0.16 

o 

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w 0.14 

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UJ 

q: 

o 

^ 0.12 
z 


^ 0.10 

o 

o 

_l 

llJ 

> 0.08 

>• 

a: 

0.06 


0.04 


0.02 
































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71 C 

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X 










































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1" VENTED MO 

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X MEAN POINT OF GROUPS 

















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20 40 60 80 100 120 140 160 180 

ENTRY VELOCITY IN FT/SEC 


Figure 64. Whip/entry velocity of prototype and 
1-in. model Steel Dummies versus entry velocity where 
9e = 19°. Points represent averages of a number of ob- 
servations. 


the case on full-scale launchings, however, as yet no 
data on this point exist. It is observed that in the 
1-in. model launchings the cavity appears to persist 
very much longer than what is believed, and some- 
times seen, to occur on the full scale. In 1-in. vented 
model launchings, the cavity is sometimes seen to 
persist even to the broach of the model. The reason 
for this difference has not been investigated by the 
Morris Dam Group. 

The trajectories after tail slap of the 1-in. Fronde 
model were seen to dive after traversing about the 




106 


WATER ENTRY 


first seven torpedo lengths despite the fact that the 
torpedo struck the bottom cavity wall and started to 
turn up. This is not a weight effect since the velocity 
is too large, and it differs very markedly from the 
prototype where the trajectories turn up. 

This behavior appeared to improve (differ less from 
the prototype) the larger the entry velocity. 

The cause of this behavior appears, as before, to be 
a down lift on the torpedo during the cavity stage. 
Hence the same remedy, namely, venting, was tried. 
Whereas, before, only the nose was vented to improve 
the whip, now the entire torpedo was vented, as 
illustrated in Figure 65. The result was a marked im- 
provement in the subsequent underwater trajectory. 
Before the rest of the dummy was vented, the model 


Figure 65. Illustration of the 2-in. vented model of 

CIT Steel Dummy. 

appeared to dive, and, after the venting, it actually 
broached. However, as in the whip, the solution was 
not complete. The trajectories depend on velocity, 
and for a Froude prototype velocity of about 300 ft 
per sec even the vented models dive while the proto- 
type continues to turn up. 

Conclusions 

The study and results of modeling indicate that 
effects which may truly be regarded as second order 
in prototype actually become first order for small, 
fine nose Froude models and hence cause very large 
discrepancies between the model and full-scale re- 
sults. 

For blunt noses or nose with discontinuities, much 
more faith can probably be put in the Froude model 
since the fine air space no longer exists. 

The modeling of fine noses has definitely been im- 
proved by venting; however, experimental results 
clearly indicate that the problem is far from a satis- 
factory and complete solution. Hence small Froude 
model results must be used with a great deal of 
caution and experienced judgment. 


6.5 RECOVERY STAGE 

6.5.1 Introduction 

During this stage the torpedo is in effectively non- 
cavitating motion and is decelerating down to its 
running speed if it is powered or decelerating down to 
zero speed if unpropelled. If the torpedo is controlled, 
it will tend to return to the set depth and direction 
for its steady run. With the ordinary controls now in 
use, if the torpedo has rolled (as it generally does), a 
hook results, but it finally heads in the direction as 
directed by the gyro. 

During this stage we know the equations of motion, 
and their solution is relatively simple. It appears that 
the motion of the torpedo can be predicted by the use 
of eight hydrodynamic constants together with a 
knowledge of the control mechanism. The subse- 
quent discussion is equally applicable to torpedoes 
without controls, with fixed controls which are ideal, 
or with actual control systems. The specialization to 
ideal and actual controls will be given a little later. 

6.5.2 Assumptions and Equations of Motion 

Assumptions 

In the general case, the torpedo is at a roll angle 
0(s) and possesses a rolling angular velocity so that, 
at the beginning of this stage of the motion, the 
orientation about its longitudinal axis may be de- 
scribed by some roll angle and angular velocity. 
During this stage of the motion the controls, if they 
are operating normally, will call for hard up elevators 
since the torpedo is well below set depth. If the con- 
trol includes a pendulum, it will be against the for- 
ward stop due to the deceleration of the torpedo, and 
this also will call for an up elevator. The vertical 
rudders, during the earlier part of this stage, will be 
hard over in a direction to prevent a change in head- 
ing due to the effect of the elevators in the horizontal 
plane that is associated with the roll displacement 
0(s) and due to the initial heading error (at the be- 
ginning of this stage) arising from the fact that the 
torpedo may have gone to one side of the cavity in 
the previous stage and may therefore be pursuing a 
curved path. Clearly, if the torpedo is without con- 
trols (like rockets and the Steel Dummy), the effect 
of roll on the motion during this stage will be prac- 
tically negligible. In view of the fact that the ele- 
vators are hard up, at least until the torpedo reaches 



RECOVERY STAGE 


107 


a depth, inclination, and speed at which the controls 
call for a different elevator angle, it may be assumed 
that during the earlier stages of the motion the ele- 
vator angle ^ has a constant value corresponding 
to the limit of the up elevator throw or = 0 in the 
particular case that the torpedo has no controls. Simi- 
larly, the vertical rudder angle 8 has the constant 
value do corresponding to the maximum rudder throw. 

In Section 7.2 the equations of motion are derived 
for a torpedo in the steady state, which is right side 
up, {(f) =0). Strictly speaking, the equations are de- 
rived either for a torpedo for which 0 = constant = 
0, 7r/2, TT, Stt 2. The general case is to be considered 
here of a torpedo at a roll angle 0(s) which is not 
constant. For a first approximation the roll will be 
introduced as an independent effect, assuming that 
the roll influences the motion of the torpedo only 
through the changing components of the elevator and 
rudder forces and moments. This assumption is prob- 
ably accurate enough for a first order theory, pro- 
vided the torpedo is somewhat symmetrical. Gener- 
ally, torpedoes are composed of a body which is a 
solid of revolution with perhaps an appendage which 
is rotationally symmetrical like the shroud ring so 
that the body has an infinite number of axes of sym- 
metry. To this body are generally added the fins and 
controls. It is the latter two appendages which, if 
they are present, cause the torpedo to deviate from 
a body of revolution. For a torpedo with four fins 
which are ecpial in area, the torpedo appears the 
same when rotated through 90°, 180°, 270°, and 360° 
angles; in other words, the torpedo then has a four- 
fold axis of symmetry. With controls there is gener- 
ally only a twofold axis of symmetry. In the assump- 
tion mentioned above it is assumed that, for example, 
in the horizontal plane the hydrodynamic coefficients 
of the torpedo (excluding controls) are to the first 
approximation insensitive to changes in roll orienta- 
tion 0. The error in this assumption is not very great 
and certainly decreases with increasing rotational 
symmetry. Thus, in this first order approach, the 
equations developed in Chapter 7 may be used if the 
effect of both elevators and rudders in the vertical 
and horizontal planes is considered. For example, for 
the elevator against the bottom stop and the vertical 
rudder against the side stop, the lift coefficient in the 
vertical plane when the torpedo is rolled through an 
angle 0 would be given by 

— ► — Cx^^o cos 0 — C\8o sin 0 . 


In Chapter 7 no account is taken of changes in 
speed of the torpedo since only the steady running 
state is considered where speed changes are negligible. 
As a matter of fact, even the very large deceleration 
during the recovery stage affects the motion only as 
it affects the control mechanism. This is true because 
of the assumption that all of the forces are propor- 
tional to the square of the instantaneous velocity and 
independent of the acceleration. 

The only exception is the case of the gravitational 
forces and moments. These lead to the terms 

2(TF - B) , 2aB 

and 

pA F2 pA F2 

so that they become more important as the velocity 
diminishes. 

If the thrust of the propellers were independent 
of the velocity it would be possible to write 

.1/,F V' = -Cd — (F- - Vr-) , (53) 

2 

where IT is the running speed. Some observations 
have led to the value of Cd = 0.45 based on this 
equation. This is obviously too large a value to repre- 
sent a true drag coefficient and it must be concluded 
that equation (53) does not really represent the true 
situation. 

That equation (53) is incorrect is really quite 
understandable for the thrust of a propeller depends 
upon its motion through the water. If the torpedo 
is moving faster than the speed corresponding to the 
propeller rotation, the propeller might even produce 
a drag instead of a thrust. A good estimate is prob- 
ably obtained by neglecting the propeller altogether 
and setting 

MiVV = -CdA-V'- (54) 

2 

as long as F > Vr and treating V as constant after it 
gets near the value Vr. Then 

2 (IF - B) ^ 2{W-B) 
pA F^ pA T V 

until 

Co^4s TV 

e = — . 

IV 




108 


WATER ENTRY 


Equations of Motion 

With all these considerations the equations of mo- 
tion in Chapter 7 when the torpedo is rolled through 
an angle 0(s) and decelerating become: 

Vertical Plane. 


since it depends on the roll velocity at the beginning 
of this stage and on the damping in roll of the 
torpedo. 

In order for equations (56 and 57) to be amenable to 
analytic integration </> must be expressed as a linear 
function of s. If this cannot be done, the equations 
may be integrated numerically. The fact that with 


m9. m <iOL + Cia = — Cx,^o cos </> — C\h^ sin </> 


p.r 


+ 


2(ir - 

0*4 TV 


(56) 


nO' + Ck9 + CmOL = cos 0 + sin 0 


+ 


A 


(57) 


where it has been assumed that cos 6=1. 


Horizonial Plane. 

(essentially replacing iS^0) 

inu) + + Cl^p = Cx,^o sin 0 — Cx,5o cos 0 , (58) 

nui' + Ca'w “h Cm^P = sin 0 -|- cos 0 . (59) 


Roll, 0(s). 

In order to obtain a solution of these equations, 
it is necessary to know how 0 varies with s. To 
obtain this dependence it appears that one must 
revert to experiment. In the case of the Mark 13 tor- 
pedo with a shroud ring, the torpedo roll appears to 
attenuate exponentially from a maximum value at 
the beginning of this stage of motion. Without the 
shroud ring the roll continues to increase for a dis- 
tance of about 250 ft and then begins to damp out. 
A possible explanation for this behavior is that the 
shroud ring increases the effectiveness of the fins so 
that they are able to damp out the motion more 
rapidly. A typical curve of roll angle versus distance 
is given in Figure 66 for the Mark 13 torpedo with a 
shroud ring when entering right side up and upside 
down. This is the second effect of the shroud ring on 
the Mark 13 torpedo that has been encountered. The 
first effect discussed in Section 6.4 was to increase the 
tail lift and hence to increase the radius of curvature 
of the path in the cavity stage and to decrease the 
mean pitch and yaw angles. For torpedoes without 
controls, no knowledge of the roll is necessary in this 
approximation. In general, the roll angle as a func- 
tion of distance must be obtained experimentally 



Figure 66. Roll of Alark 13 torpedo with shroud ring 
versus distance from entry. 


the shroud ring the roll seems to attenuate exponen- 
tially indicates that the motion is over-damped. 
Hence it might be expected that 0 = 0oC~“k Then 
as this expression is expanded and two terms retained 
thus 

0 = 00 (1 — as), 

where 0o is the roll angle at the beginning of this 
stage and a is obtained from the slope of the curve of 
0(s) in Figure 66 (s is in torpedo lengths). It appears 
that for 00 = 90°, a0o = —12.5 degrees/torpedo 
length and for 0o = 150°, a0o = —21.5 degrees/tor- 
pedo length, approximately a = 0.14 per torpedo 
length. 

This approximation must be treated somewhat 
differently without a shroud ring since there is a 
very small attenuation of roll. Clearly, it is also ap- 
plicable to torpedoes without controls. 


RECOVERY STAGE 


109 


Then, writing 

1 = ^X^sO cos <f>0 , 

- 1 2 = T sin 00 , 

^■1 3 = cos 00 , 

Ai = C sin 00 , 
the equations of motion 


^1 1' = Cx, 5 o cos 00 , 
A o' = Cx^do sin 00 , 
As' = C fi^do cos 00 , 
A 4' = C^ 8 o sin 00 , 

become : 


ment that a necessary and sufficient condition for 
the equations, 

77i9.i + (niopi + Ci)ai = 0 , 

{fiPi + CK)9i -f- CmOii = 0 , 

to have a nonzero solution is that the determinant of 
the coefficients of a,; and 9 i must vanish. Hence 


Vertical Plane. 

+ Cia = — (Hi + A o') cos (as0o) 

r 

2 fTT" — B)e 

(Ho - Ho') sin (as0o) + , (60) 

pH T 0“ 

n9' Ck9. + Cm(x = (H3 -j- H4') cos (as0o) 

+ (H4 — H3') sin (as0o) + ^ . (61) 

pH T 0“ 

Horizontal Plane. 

mo) + = (-“12 — Hi') cos (as0o) 

— (Hi + H2') sin (as0o), (62) 

72 co' + Ca'co + C^0 = — (H4 — H3') cos (as0o) 

+ (H3 + H4') sin (as0o). (63) 

These equations are integrated up to a value of s 
such that 0 = 0, at which distance the torpedo is 
right side up. Since the motion is over-damped, from 
that position onwards 0 0. Hence at the value of s 

such that 0(s) =0 the new equations of motion are 
dealt with, which are the same as the ones up to this 
point, except that it is assumed 0 = 0 on the right 
side of equations (56) to (59). 

Solution of Equations of Motion 
The general solution of the equation is: 

Vertical Plane. 

9. = QiC^i* + 9.oe^^^ + 9s cos (0oas) + 9.4 sin (0oas) 

+ 9se''^ -f- 9^ , 

a = aicPi® + + «3 COS (0oas) + 0:4 sin (0oas) 

-j- 0:56* + ae • 

The value of pi and p2 is determined by the require- 


Pi = 

_ (pG-|-??Z2CA)dz\/ {nCl + 7n2CKy~ — Vl77l2{ClCK — ^^^Cm) 

2nm2 

a = 1,2). 

The constants ai, a2, 9i, and 92 are determined from 
the initial conditions of the motion, a* and 9* = l/R 
as given in Section 6.4, and also by the equations of 
motion (56 and 57) setting s = 0. Thus a* and 12* 
with the equations of motion (60 and 61) at s = 0 de- 
termine «!, 0:2, and 122. Thus there are really only 
two independent constants since the other constants 
0:3, «4, «5, 123, 124, 125, and 126, are completely deter- 

mined. Equating coefficients of cos (0oas) and sin 
(0oas) one obtains four equations with the four un- 
knowns as, 0:4, 123, and I24 xvhich may easily be solved, 
thus determining as, 0:4, I23, and I24. In a similar man- 
ner, q ;5 and 125 may be determined from the equations 
of motion simply by equating coefficients of e''\ The 
constants a^ and 126 are determined by equating co- 
efficients of the constant terms in the equations of mo- 
tion. When 0(s) = 0, the equations of motion become 

7n9 + 7n2a' -f Cia = — (Cx^o — Wo) —lOoPP'^ , 

7i9' + Ck9 -j- Cma = ((7;u^o + 5o) — h^F e '^^ , 

where 

2 (IE - B) 

IVo = , 

pHEn(0 = O)] 

^ _ 2Ba 

pHFn(0 = O)] ’ 

and (Cx^o — Wq) and (C^u^o + 5o) are constants. 
E“[(0 = 0)] means V^ evaluated at the value of s 
where 0 = 0. 

Then the general solution is 

a = aiC^i® 0:26^2® asC^ -f- 0:4 , 

12 = 12iC^^i® -f" 1226^*2® -|- 123C® -[- 124 . 


110 


WATER ENTRY 


The constants of integration may be obtained as out- 
lined earlier. 

Then, since 6 = a (3, the change in depth is given 
by 


An actual control system usually includes a pen- 
dulum, and in a system similar to that in the Mark 13 
torpedo 

^ — r(/3 -f- A^) — ah , 


D = Do -Csi 


sin (a + ^)ds~Dc 


-I 


s(0 = 0) 


-f 


s{4> = 0) 


{a + ^)ds 
(q! T" /3)ds , 


X = .To COS {d -h / 3 )f/s ~ :ro + s , 

where /3(s) = do depth at the 

beginning of this stage of the motion (or at the end 
of the transition stage as given under Condition of the 
Torpedo When Effectively in N oncavitating Motion in 
Section 6.4.4. do may also be obtained from Condition 
of the Torpedo When Effectively in N oncavitating Mo- 
tion, Section 6.4.4. Thus do = d* — a*. 

It has been assumed that {a + d) is usually small 
enough so that sin (a d) = a -f- d- This is suffi- 
ciently accurate for upturning trajectories and some 
downturning trajectories. For most torpedoes going 
to the top of the cavity the trajectory angle becomes 
large so that (a + jS) > > sin (a-\- d)- The integration 
can be carried through graphically or numerically in 
such a case. 

In the horizontal plane there is a completely analo- 
gous solution, except that db = = do = 

becomes the course deviation in the horizontal plane. 

It is to be noted that in the special case where the 
torpedo does not have controls Cx^o = Cf,^Q = C\8 q = 
C^8o = 0 and the equations of motion are very much 
simplified. 

Controls 

Torpedoes with controls possess a control system 
that transmits signals to the elevators and rudders. 
The signals transmitted depend on the orientation 
and the position of the torpedo. Two types of control 
systems may be considered, an ideal one and one 
similar to that of the Mark 13 torpedo. 

AVe shall take as an ideal system, one in which 


^ = kr-\d- Clh , 


where is the steady running elevator angle, X and a 
are constants depending on the control. 


where X is the angle the pendulum makes in space, 
so that — (d + A^) is the angle the pendulum makes 
with the torpedo axis. This relationship between the 
attitude of the torpedo and the elevator position is 
correct only in the case of steady motion. A decelera- 
tion will change the reference direction and produce 
a spurious response. Although both of these expres- 
sions refer to controls of the proportional type, simi- 
lar considerations are also applicable to on-off (two- 
position) controls since they behave like proportional 
controls in many respects (see Section 8.4). 



Figure 67. Theoretical trajectories for CIT Steel 
Dummy for various pitch angles at entry. 


In the ideal case, the controls govern the elevators 
when the value of ^ called for by d and h is not greater 
than ^ 0 - Similarly, the controls of the Mark 13 tor- 
pedo take over when d and X have the values such 
that 1^1 ^ ^ 0 . This may be obtained from the equa- 
tion of motion of the pendulum, equation (1) in 
Chapter 12. 

At this point along the trajectory the ecpiations of 
motion given in Chapter 7 are used, and the value of ^ 
as given by the control equations is inserted in them. 
Then the set of differential equations is readily inte- 
grated. 

In the horizontal plane, the controls of the Mark 13 
torpedo are of the on-off type. Hence the vertical 
rudder is usually over against the stop until the angle 
between the torpedo axis and the gyroscope becomes 
zero. 


RECOVERY STAGE 


111 


As an example of the calculation of the trajectory 
for a torpedo without controls the equations of mo- 
tion in the vertical and horizontal plane for the CIT 
Steel Dummy can be solved. 

In the vertical plane, the theoretical trajectories 
for de = 20° are drawn for various pitch angles at 
entiy in Figure 67. One case is included where the 
pitch angle at entry is more nose-down than the 



— NOSE DOWN NOSE UP ► 

PITCH ANGLE AT ENTRY, oie 'N DEGREES 

Figure 68. Theoretical maximum depth of dive 
versus pitch angle at entry. 


critical pitch angle. These trajectories are in close 
agreement with observed trajectories at the CIT- 
TLR. In Figure 68 the theoretical maximum depth 
of dive for Be = 20°, = 0°, is plotted against the 

entry pitch angle. From this figure the dependence of 
the maximum depth of dive on entry velocity is seen 
to be very slight. 

In the horizontal plane, by theory it is readily 
found that for i/'e = 3.5°, ae = —2°, the lateral course 
deflection of the Steel Dummy at 550 ft from entry 
is about 40 ft, which is very close to the observed re- 
sults. In Figure 69 a graph is plotted of the theoretical 
and observed results for the angle in the horizontal 


plane between the torpedo axis and the direction of 
launching. A fair agreement with experiment is noted. 

By methods outlined, similar calculations can be 
carried out for torpedoes with controls whose hydro- 
dynamic constants are known. If the dependence of 
the entry whip on the trajectory angle is known, it is 
a simple matter to calculate the depth of dive as a 
function also on the trajectory angle at entry. In the 
absence of precise knowledge of this dependence, it 
could probably be assumed that the variation is as 
the cot Be for which there is some evidence. 


6.5.3 Description of Motion in Recovery Stage 
General Behavior 

Generally, the torpedo begins the stage with a large 
pitch angle (nose-up or nose-down) and a yaw angle. 
Until these are reduced to small values so that the 



Figure 69. Angle in horizontal plane between axis of 
Steel Dummy and direction of launching. 


torpedo is ‘‘on its trajectory,” large lift and cross 
forces are experienced. Thus, if the torpedo is on the 
bottom of the cavity, the large initial nose-up pitch 
angle produces a lift force that counteracts the effect 
of the controls even if the torpedo is upside down. If 
the torpedo is at the top of the cavity, even if it is 
right side up, the forces due to the nose-down pitch 
continue the torpedo on its downturning trajectory, 
causing a great depth of dive. Thus there is a transi- 
tion region during which the torpedo settles down to a 
steady motion, the transients damp out, and the tor- 
pedo tends to line up with its trajectory. 

In the horizontal plane, since the torpedo possesses 
an initial yaw angle and angular velocity and also 
since the torpedo is generally rolled over initially. 




112 


WATER ENTRY 


there are forces tending to produce a course deflec- 
tion, and the path curves away from the straight line 
in which the torpedo was aimed. Clearly, this deflect- 
ing force and moment diminish for two reasons as the 
torpedo proceeds along its trajectory. First, the yaw 
angle and hence the cross force diminishes; and, 
secondly, the torpedo is righting itself, and the effect 
of the elevators in producing a course deflection di- 
minishes as the torpedo rights itself, while the effect 
of the rudders in straightening the path continues to 
increase. As a result of these effects the path curves 
toward the desired direction until it finally straight- 
ens out. The net result is a lateral displacement of 
of the path of the torpedo. 

The effect of the controls in producing a hook is 
probably considerably greater than the effect of the 
initial yaw angle ^pe that causes the torpedo to ride on 
one side of the cavity. The effect of the elevators in 
producing this hook is indicated by observations at 
the CIT-TLR. On most of the launchings the tor- 
pedo is rolled at the beginning of this stage of the 
motion between 0 and — 180° (a counterclockwise 
roll). As a result a left hook is expected. However, 
this hook has been considerably diminished, and 
even a right hook has been produced by means of the 
counter pendulum and anti-pendulum. By these de- 
vices the elevators are held in a neutral or down po- 
sition for a short time after entry. 

Effect of Shroud Ring 

During the open cavity stage it was seen that the 
shroud ring produced an additional lift at the tail and 
thus increased the radius of curvature of the trajec- 
tory. This effect tends to create a greater depth of 
dive for the torpedoes on the bottom of the cavity 
and a shallower depth of dive for torpedoes on the 
top of the cavity than is the case of a torpedo that has 
no shroud ring. It also tends to produce a smaller 
hook for torpedoes striking the side cavity wall. 

The principal effect of the shroud ring probably 
takes place after cavity collapse and may be traced 
to the decrease in the effectiveness of the elevators 
and rudders, the increase of the static stability of the 
jMark 13 torpedo, and the increased damping of roll 
which it produces. The first effects combine to in- 
crease the turning radius of the torpedo both in the 
horizontal and vertical planes by roughly a factor of 
4 to 5. As a result, the large hook effected by a tor- 
pedo without the shroud ring is considerably dimin- 
ished since the torpedo turns in a much larger circle. 
Some tests were run with the shroud ring in a posi- 


tion somewhat farther forward on the fins than the 
standard position. On the Mark 13 torpedo the for- 
ward position of the shroud ring gives the controls an 
intermediate effect and hence an intermediate turn- 
ing radius, and probably does not damp out the roll 
as rapidly as the aft position. Hence an intermediate 
value in the number and extent of the hooks for the 
forward position may be effected. All these conclu- 
sions have been observed in full-scale launchings at 
various naval torpedo stations. 

The effect of the shroud ring on the depth of dive 
may also be explained by the differences produced in 
the hydrodynamic constants of the torpedo. The 
turning circle in the vertical plane is of considerably 
larger radius with the shroud ring aft than without 
the shroud ring and is roughly intermediate between 
these two values for the shroud ring forward. Hence, 
for a given set of conditions at the beginning of this 
stage of the motion, the torpedo without the shroud 
ring will tend to turn up in a very tight circle (or turn 
down if it is rolled over), while with the ring it will 
turn up in a comparatively large circle. As a result, 
it is expected that the depth of dive without the ring 
will be less than with the ring aft and the depth of 
dive with the ring forward will be in between. These 
conclusions have also been verified in full-scale tests 
at various naval torpedo stations. 

Broaching 

Alany torpedoes emerge from the water after 
traveling some distance beneath the surface. This is 
called a broach. The cause of broaching lies in the 
trajectory during the open cavity stage as well as in 
the torpedo controls. Clearly, if the cavity does not 
close until the torpedo has progressed some distance 
under water and if the torpedo strikes the bottom side 
of the cavity, it will pursue an upturning trajectory 
which has been shown to be roughly circular in form. 
Consequently, the torpedo may emerge from the 
water while in this open cavity stage, as appears to 
be the case in very high speed rockets. In addition, 
even if the cavity closes while the torpedo is still sub- 
merged but the torpedo has turned up sufficiently so 
that it is headed toward the surface of the water, it 
may also broach. Thus it may be said that broaching 
will depend on the trajectory angle at entry Be since 
at cavity closure 6 = Be — M and AB is approximately 
constant. In addition, it is necessary for the torpedo 
to be on the bottom side of the cavity if it is to pursue 
an upturning trajectory; hence broaching will depend 
on whether ae > ac or ae < ac. Furthermore, the 












RECOVERY STAGE 


113 


more nose-iip ae the shorter will be the distance to 
tail slap, aS, and the less will be 6 at cavity closure. 
Broaching will also depend somewhat on Ye since the 
greater Ye the longer the time to cavity closure and 
hence the smaller 6, with a resulting greater tendency 
to broach. These conclusions have been verified in 
model and full-scale tests of bombs. Also, most 
rockets broach, and this mav be attributed to small 
de and large 

In the case of torpedoes with controls there is an 
additional effect. It has been seen that at the begin- 
ning of the stage where the motion is in noncavitat- 
ing water the torpedo is below set depth and deceler- 
ating, with the elevators, as a result, hard up. Even 
when the torpedo has reached its set depth it may 
still be decelerating sufficiently that the elevators 


remain hard up with the torpedo coming out of the 
water. It is expected that this phenomenon is more 
prevalent the higher Ye and has been so observed in 
British torpedo launchings. 

When the torpedo broaches, its heading may 
change as it leaves the water. The torpedo may hook 
in either direction and it also may not rotate about a 
transverse axis in air with the result that it lands 
tail first. In this instance, there is another effect of 
the shroud ring. Due to the ring there is a restoring 
moment at the tail of the torpedo; when the nose 
emerges this moment tends to give the torpedo a 
nose-down angular velocity so that it re-enters head 
first. In addition, this moment tends to keep the tor- 
pedo on course so that the hook is small at broach- 
ing. 



Chapter 7 

UNDERWATER RUN 


7.1 ESTIMATES OF COEFFICIENTS 

V ALUES OF THE drag coefficient Cd, the moment 
coefficient Cm, and the lateral force coefficient 
Cl, defined in equations (3), (1), and (2) of Chapter 
4, can be determined from model tests or from tow- 
ing, wind tunnel, or running tests with the full-scale 
torpedo. Because of the inaccuracy inherent in scal- 
ing up the drag from model to prototype, Cd would 
probably best be determined from a full-scale run in 
which the propeller thrust was measured by apparatus 
contained in the torpedo. In practice, however, model 
measurements are much easier to perform, and an 
extrapolation of Cd determined in the high-speed 
water tunnel as a function of Reynolds number is 
actually used. 

The moment and force coefficients have been 
measured using 1/11-scale models in the high-speed 
water tunnel at the California Institute of Tech- 
nology and 1 /5-scale models in the wind tunnel at 
Gould Island. Values obtained in these two tunnels 
are not in complete agreement, and it appears that 
the discrepancies increase as the l/d ratio. They are 
probably due primarily to the smallness of the par- 
ticular forces and moments being measured and to 
the difficulty of correcting for interference between 
model and support shield. 

The moment increases from zero as the attack 
angle increases from zero, linearly at first, and less 
rapidly than this for larger angles. It has the sign of 
an upsetting moment, tending to increase the attack 
angle for small values of this quantity. It is because 
of this that torpedoes are generally regarded as being 
statically unstable; however, as will be shown in 
Section 7.2, this is largely irrelevant to the actual 
dynamic behavior of the torpedo in the water. The 
lateral force also increases linearly with attack angle 
for small angles and somewhat faster than linearly 
for larger angles. It has the same sign as the lift of an 
airfoil section: the torpedo tends to translate in the 
direction of the deflection of its nose. Typical curves 
of Cd, and of Cm and Cl as functions of attack angle 
for 0° and 2° elevator (horizontal rudder) angle are 
shown in Figure 1 . These are rough provisional values 
for the Mark 13-2 torpedo, with and without the 8° 
cone angle shroud ring in the aft position; the vertical 


rudders are about one-tenth as effective as the ele- 
vators. The overall length of this torpedo is I = 13.42 
ft, the maximum cross-sectional area is A = 2.75 
sq ft, and the density of sea water is p = 2 slugs per 
cu ft. 

In addition, it is often useful to remember that, 
since the center of pressure of the elevator lift is close 
to the elevator stock, the ratio of the elevator mo- 
ment coefficient {C,,) divided by the elevator lift 
coefficient (Cx) is given by X 7, where X is the dis- 
tance from center of gravity to the elevator stock and 
I is the torpedo length. Since for many torpedoes (for 
example, the Mark 13-2) X = / 2; C^/Cx = 0.5. 
Similar considerations hold for the rudder coeffi- 
cients. 

The damping moment and force coefficients Ck and 
C F are much more difficult to estimate since the tor- 
pedo or model must be rotating as well as translating 
through the water. They could, in principle, be de- 
termined experimentally from oscillating models in 
the water tunnel, from straight models in a curved 
tunnel or curved models in a straight tunnel, from 
rotating arm or curved rail towing tests with model 
or prototype, or from free turning tests with pow- 
ered models or full-scale torpedoes. The last is the 
only method that has been used ; photographs of cir- 
cular runs have been made at the California Insti- 
tute of Technology, and depth and roll records have 
been collected from figure running torpedoes at New- 
port. It is indicated in Section 7.2 how a relation be- 
tween Ck and C f may be obtained from the measured 
turning radius with known rudder angle; it is, of 
course, impossible to determine both of these un- 
knowns from a single piece of data. 

In this connection further information may be ob- 
tained from a theoretical calculation based on the 
moments and lateral forces measured on models of 
the bare hull and the complete torpedo. It is assumed 
that the damping moment and force arise principally 
from the tail empennage and are due to the fact that 
the attack angle at the tail differs from that at the 
center of gravity by coX ' radians, where X is the dis- 
tance from the center of gravity to the center of 
pressure of the empennage (see Section 4.4). The de- 
pendence of empennage force on attack angle and the 
magnitude of X can be inferred from the difference 




114 




ESTIMATES OF COEFFICIENTS 


115 


0,06 


0.05 


0.04 


'M 


0.03 


0.02 


0.01 






2‘ 

[7^ 

/ 



. — 

— — . 





/ 

/ 

A 

/ 

b- 






/ 

/ 

r 

/ 

/ 








/ 

/ 

i 

/ 

/ 








/ 

f 

/ 

/ 








/ 

/ 

/ 

/ 

/ 






> 


! 

7 

t 

T 





0 



/ 

/ 

/ 

/ 








t 

/ 

' / 

/ 








/ 

/ 

/ . 

/ 









/ 

! > 

t / 




SHROUD TAIL 

--PLAIN TAIL 


/t 

/ 






1/ 












0 1 2 3 4 5 6 7 .8 9 10 

ATTACK ANGLE IN DEGREES 

Figure 1. Cd, Cm, and Cl as functions of attack angle for 0° and 2° elevator (horizontal rudder) angle. 


between the forces and moments experienced by the 
complete torpedo and the bare hull; from these, 
values of Ck and CV are readily estimated. The 
British Torpedo ^Manual (p. 122 of the 1929 edition) 
recommends that Ck be increased slightly beyond the 
value obtained in this way to take account of the hull 
damping moment. The following provisional numer- 
ical values for the Mark 13-2 torpedo were arrived at 
by considering both the experimental and theoretical 
approaches: for the plain tail, Ck = 0.44, Cf = 1.01; 
for the shroud ring tail, Ck = 0.48, Cf = 1-1 6. Inci- 
dentally, it appears on the basis of this discussion 
that for conventional torpedo shapes Ck and Cf may 
be related to Cm/ and Cl/, where the subscript / 
refers to the corresponding coefficient for the fins. 

F^p to the present, practically all static testing, like 
the tests run in the water tunnel and wind tunnel 
were made on torpedoes without propellers. It is a 
well-known fact from other fields (i.e., airplanes, 
ships, etc.) that the propellers exert a side force. As 
a result they create some stabilizing effect (dC.u/da 
is diminished), the propellers produce a lift force, 
and they contribute very significantly to Ck and Cf. 
Hence the omission of propellers probably leads to 
large errors in the various constants, ('m, Cl, Ck, and 
Cf. Although Ck and C f have been determined pri- 
marily from full-scale turning results, the relation be- 


tween Ck and Cf depends on Cm and Cl which are 
certainly in error. Recently, in the Gould Island wind 
tunnel the Mark 13-2 torpedo was tested with free- 
wheeling (windmilling) propellers. It is as yet not 
known whether results obtained with freewheeling 
propellers may be taken as the results obtained with 
powered propellers, even though both are rotating 
at close to the same speed. However, the observed 
results, which are presented below, are indicative of 
the effect of propellers on the static constants. 



dCM 

dC L 


da 

da 

Bare hull 

0.952 

1.16 

Bare hull plus free props 

0.796 

1.46 

Bare hull plus plain tail 

0.640 

1.80 

Bare hull plus plain tail jilus jirojis 

0.466 

2.14 

Bare hull i)lus shroud ring tail 

0.419 

2.26 

Bare hull plus shroud ring tail plus props 

0.36:3 

2.36 


From these results it appears that in addition to 
the large effect of propellers there is an interference 
effect with the shroud ring and propellers. These 
results are only preliminary, and it is to be remem- 
bered that the propellers are freewheeling. 

It thus appears that the status of the hydrody- 
namic constants of torpedoes is at present very un- 
satisfactory and the knowledge inadequate. The 


116 


UNDERWATER RUN 


static constants Cm and Cl are in error since the tor- 
pedo models have been tested without propellers. 
The dynamic constants Ck and Cf are in error since 
the wrong static constants were used in their calcu- 
lation, the turning circle data must be corrected for 
roll (as will be seen in Section 7.2). In addition, the 
turning circles very often are not reproducible. It 
appears that future work in determining these con- 
stants by means of models with propellers (prefer- 
ably powered at least to see the limits of freewheeling 
propellers) as well as curved motion tests is definitely 
required. In fact, it appears that results of this type 
may soon be obtained in towing and rotating arm 
tests. 

7.2 EQUATIONS OF MOTION AND 

STABILITY 

In studying the dynamic behavior of a torpedo, it 
is convenient to separate the motion into components 
in a horizontal plane, in a vertical plane, and about 
the longitudinal axis of the torpedo. This is a natural 
separation from the point of view of the control sys- 
tem, for the steering, depth and heel are handled in- 
dependently. In general, the interaction between 
them is small (see Section 8.5), and the approxima- 
tion is both valid and useful. The equations are de- 
veloped here for motion in a vertical plane (depth- 
keeping); however, they are readily adapted to the 
steering motion in the horizontal plane. Rolling and 
heeling will be considered in Chapter 9. 

The three degrees of freedom in the vertical plane 
are best taken as the components of the motion of 
the center of gravity along and perpendicular to the 
longitudinal axis of the torpedo and the angle of ro- 
tation about a horizontal transverse axis through the 
center of gravity. The choice of axes fixed in the tor- 
pedo rather than in space or with respect to the tra- 
jectory of the center of gravity has the advantage 
that accelerations along and perpendicular to the 
axis of the torpedo are treated separately so that the 
accession to the inertia of the torpedo due to en- 
trained water, which is different for the two direc- 
tions, can be readily taken into account. The equa- 
tions of motion are 

Mill — ]M‘iV& 

= P — D cos a + L sin a + (IT — R) sin , (1) 

JM 2^ JM ill I3 

= —L cos a — D am a (IT — B) cos /3 + (2) 


Q(3 = M — Ba cos jS , (3) 

where the left sides are the inertia terms given by 
Lamb"^ and the right sides are the hydrod^mamic 
forces and moments; Lamb’s term (il /2 — Mi)iiv is in- 
cluded in the measured moment M, and terms arising 
from fore-and-aft asymmetry are neglected. Here, 
Ml and M 2 are the mass in slugs of the torpedo cor- 
rected for entrained water in longitudinal and trans- 
verse accelerations, respectively; Q is the similarly 
corrected moment of inertia in slug-ft-; u and v are 
the longitudinal and transverse components of the 
velocity V in ft per sec; (3 is the orientation angle, a 
the attack angle, and 6 = ^ a the course or trajec- 
tory angle, all in radians; IT is the weight, B the 
buoyancy and P the propeller thrust, all in lb, and 
the remaining forces and moments are measured in 
lb and ft units; ^ is the elevator angle in radians and 
a the distance of the center of buoyancy (CB) aft of 
the CG in ft; dots represent time derivatives. 



Figure 2. Diagram showing positive direction for 
various parameters. 

Figure 2 illustrates the sense in which the various 
parameters are taken positive. M and L are, of 
course, functions of a and p, however, D appears to 
be practically independent of these quantities (see 
Figure 1). 

For motion in the horizontal plane, equations of 
the above form are valid, except that the terms in- 
volving IT and B no longer appear. Then equations 
(2) and (3) may be applied to steady turning, if u, 
i), and (3 are set equal to zero. It is then apparent that 
the turning radius is F/jS so that (3 as well as ^ is 
measurable. Then a can be eliminated from the two 
equations and a relation obtained between K and F; 
this is the relation referred to in Section 7.1. In inter- 
preting this relation a correction should be applied 
for the heeling of the torpedo in the turn; however. 

Hydrodynamics, H. Lamb, Fourth Edition, 1916, Chajiter 
VI, pages 124, 127. 


EQUATIONS OF MOTION AND STABILITY 


117 


this correction is both small and of uncertain magni- 
tude and has generally been neglected. 

In order to treat types of motion in Avhich a, /?, 
and are small, it is convenient to linearize equa- 
tions (1), (2), and (3); indeed, it is only in this way 
that analytic solutions may be obtained. One can 
then put cos = cos a = 1, sin = j8, sin a = a, 
u = y , V = Va, M = —ca-{-d^,L = ea-\- f^, and 
assume that P, D, F, and K are independent of ^ and 
Then to first order in the small quantities jS, /S, a, 
V, and ir — B, these equations become 

M,V = P-D, (4) 

3 / 2 ! dTd/iT (3 = —€a — — ilQ;T(II — B)~\~F^) (5) 
Qd = — Coe T — P (3 Bq . (6) 


In analogy with equations (I), (2), and (3) it is 
convenient to define moment and force coefficient 
derivatives as follows: 


D e 

c 


= ipr^ACi, 


f = ipr^ACx, 

d = ypVKMC , , 



so that 


equations (5) and (G) become to first order 

+ Cia + = —C\^ + w , 

CmOL H~ T Ca'H — C T b • (10) 

Equation (4) is not of interest for the case in which 
the speed is constant; it was used in Section 6.5, 
in connection with deceleration during the initial 
dive. 

For motion in the horizontal plane the same de- 
velopment of the equations of motion follows as in 
the vertical plane except that the terms involving 
w and b no longer appear. 

An important feature of equations (10) is that 
neither the t nor the speed V appear explicitly except 
for the speed-dependent terms w and h. Thus in the 
horizontal plane (in which w and h do not appear) 
the space trajectory described Avith fixed rudder is 
independent of the speed, even if the speed is not 
constant. In particular it is of interest to consider the 
motion in the horizontal plane AA’ith neutral rudder 
(^ = 0). This provides a criterion for dynamic sta- 
bility of the torpedo itself before the control system 
is added. 

The equations to be solved are 


C L A~ C DOi — C la A~ , Cm — —Cr7,c^ T . (8) 

One can also define a dimensionless independent 
variable s such that dt = {I/V)ds is the time required 
for the torpedo to traATl ds lengths at the instan- 
taneous speed V; then 

da ds da V . , 

a = — = — — = — a , etc., 

dt dt ds I 


'm^a -|- Cia -j- — 0 , 

Cjna -j- 71^1' -)- Ck^ ~ d j (f 1) 

since these are homogeneous linear differential eejua- 
tions AAuth constant coefficients, the only non-vanish- 
ing solutions for a and 12 are of exponential form : 

a = 4- a-zC ^^^ , 

Q = -f- ^ (22) 


Avhere primes represent derivatives Avith respect to s. 
With equation (4) and the additional substitutions: 


7ni = 


24 / 1 

17’ 


7712 = 


24/2 


n = 


pAl^ ’ 


771 = 771 1 — Ci 


W = 


2 (IF - B) 


p- 


2]/2 


h = 


2BA 


P-' 


AIV^ 


Substitution of (12) and (11) giA^es the folloAAung 
equations for the coefficients of each of the exponen- 
tial factors: 


(771 2Pi + Ci)ai + 77l^i = 0 , 

Cmai -f- (7ipi -j- CK)^i = 0 , (13) 

Avhere i = 1, 2. Equations (13) have non-vanishing 
solutions for at and 12^ only if the determinant of their 
coefficients vanishes. This results in a quadratic 
equation for the characteristic exponents pp. 




(9) n7n.2Pi^ + (ni2CK + nC^)pi -f (CiCk — niCm) =0 (14) 


118 


UNDERWATER RUN 


which has the two solutions : 


Vh 


9 


a/^ 2 — 4nw2 {CiCk — rnCm) 

h rb 

2nm2 


(15) 


where h = jti^Ck + nCi. 

Now in order for the torpedo to be dynamically 
stable without controls, any initial disturbance (non- 
vanishing initial values of a or Q) must be damped 
out as the motion progresses; this means that the 
exponents pi and p 2 must have negative real parts. 
It is apparent that this is equivalent to the require- 
ment that the three coefficients of equation (14) 
have the same sign ; since the first two are necessarily 
positive, this means that the criterion for dynamic 
stability is 

CiCK-mCm>0. (16) 

It is worth noting that the torpedo need not be 
statically stable in order to be dynamically stable, 
and the latter requirement is of much greater sig- 
nificance so far as the behavior of the torpedo in the 
water is concerned. Indeed, all known torpedoes are 
statically unstable and are dynamically stable. As 
remarked in Section 7.1, static stability implies that 
Cm be negative so that a small attack angle produces 
a restoring moment. While according to (16) this is 
practically always a sufficient condition for dynamic 
stability, it is evidently not a necessary condition. 
The Alark 13-2 torpedo with either the plain or 
shroud ring tail is dynamically stable, even though 
Cm is positive; this has been shown both by computa- 
tion and by free running trials with rudders removed. 

Approximate values for the characteristic expo- 
nents Pi and p 2 are readily calculated from equation 
(15) with the help of the provisional values of the 
coefficients which are collected in Section 9.1. The 


results are 




Plain 

Shroud 

Vi 

-0.2.3 

-0.56 

7>2 

-2.91 

-2.72 


Since in each case p 2 is much larger in magnitude than 
Pi, the rate of recovery of the uncontrolled torpedo 
from an initial disturbance is determined principally 
by pi. This recovery is roughly exponential, the dis- 
turbance being reduced to 1 T of its initial value in 
approximately 4.3 lengths for the plain tail torpedo 
and 1.8 lengths for the shroud tail torpedo. Thus the 
addition of the shroud ring significantly increases the 


stability and also makes the control problem much 
less critical as will be seen in Part III. This improve- 
ment is due primarily to the decrease in Cm caused by 
the shroud ring. It should be remarked that recovery 
from an initial disturbance implies only that a, 
and hence ^ approach zero as the motion progresses. 
It does not, of course, mean that the new straight 
course is parallel to the initial course, and, in general, 
this is not the case; on the other hand, dynamic sta- 
bility means that a small initial disturbance does not 
cause the torpedo to wind up into a spiral or circular 
trajectory. 

As a result of this discussion it appears that a tor- 
pedo is dynamically stable if a perturbation of the 
yaw angle or angular velocity diminishes in time 
with a resulting relatively small course deviation. 
For example, if an uncontrolled torpedo (with rud- 
ders neutral) which is traveling in a straight course 
suddenly receives a small yaw angle and/or angular 
velocity, if the torpedo is dynamically stable the yaw 
angle and angular velocity will both decrease to 
zero, and the torpedo will resume a new straight 
course which is generally close to the original course. 
If the torpedo were dynamically unstable the yaw 
angle and angular velocity would both increase, the 
path tending to become a spiral, until finally due to 
nonlinearities in the hydrodynamic constants the 
torpedo becomes dynamically stable when the yaw 
angle and angular velocity become large enough and 
the trajectory winds up into a circle. For an uncon- 
trolled dynamically stable torpedo on a straight 
course, one readily finds from equations (II), (12), 
and (15) that the final course angle when the torpedo 
has an initial yaw angle ao and angular velocity Qo is 
given by 


00 ) — $Q -\- 


nCi% — m2CmOiQ 

CiCk — mCrn 


[Equation (20), 
Chapter 11] 


It is interesting to note the mechanism of dynamic 
stability. This is best done by considering a statically 
unstable torpedo which possesses, for example, an 
initial yaw angle to starboard and angular velocity to 
starboard. Due to the yaw angle a lift force is pro- 
duced in the starboard direction, thus giving the tor- 
pedo a velocity in the starboard direction, which is 
seen to be effectively some yaw angle in the port 
direction. Due to the initial angular velocity and the 
increase in this velocity due to the static instability, 
a damping moment is produced tending to diminish 
this angular velocity, and a damping force is pro- 


STEADY CIRCLING 


119 


duced which essentially helps the lift force in reduc- 
ing the initial yaw angle. Hence it is clear that the 
essential mechanism of dynamic stability is that the 
torpedo can move transversely and as a result an 
initial disturbance is reduced or compensated for by 
a generally small course deviation. From this discus- 
sion the main reason for the inadequacy and imprac- 
ticabilitv of static stabilitv as anv criterion for the 
motion is fairly obvious. The reason is that static 
stability is strictly defined for a torpedo which cannot 
move transversely, and it is precisely this motion 
which is the essence of the mechanism of d^mamic 
stability. 


7.3 STEADY RUNNING 


To obtain the steady-state (or mean) running con- 
ditions of the torpedo, we simply set all derivatives 
(remembering that 9. = equal to zero in the equa- 
tions of motion above. One then readily finds that, 
fora' = 9.' = 9. = 0, 


hC\ wC fx 
ClC, -h CmCy. 


Ba 


Cx + (Tf^ - B)C, 

I jU 


ClC n -f- CmC\ 


AT' 


, (17) 


hCl — wCm 
CiC ^ “h CmC\ 


— Cl - (Tr-B)G 


CiC, + C„,Cx pA T'2 ■ 

(18) 


From these equations it is seen that for a given 
torpedo the steady-state pitch angle and elevator 
angle become smaller as F, the running speed, in- 
creases. One also sees from these equations that for a 
given torpedo there is a minimum speed at which it 
can be run, if there is a lower stop in the elevator so 
that ^ is limited. 

From the second equation it is clear that one can 
choose the center of buoyancy and center of gravity 
in such a relative position that = 0. This may l^e of 
significance since when the torpedo is heeled over in 
a turn affects the turning circle. However, if = 0, 
there will be no effect of the elevators on the circle. If 
< 0, the elevators will aid the rudders in a turn and 
so contribute to a tighter circle. 


7.4 STEADY CIRCLING 


cling condition. Then from equations of motion in 
the horizontal plane it follows that: 


/ _ CiC, + a,rx 
R CiCu - rnCm ^ 


(19) 


where R is the radius of the turning circle. 
Also 

Cx-Cx -h mC, . 

a = I , 

CiC,-mC,n 

or 

CiCfx CmC\ 

9 = a . 

viC fx H- CkC\ 


( 20 ) 


These equations indicate a linear relation between 
9 and ^ and between 9 and a. 

However, the above treatment is really an un- 
warranted oversimplification of the problem of the 
circling of a dynamically stable torpedo because the 
heel in the turn is completely neglected. Most tor- 
pedoes in the steady-running condition normally 
have a down elevator [b is negative in equation (18)]. 
This is not necessarily true since it will depend on the 
relative position of the center of gravity and center 
of buoyancy as discussed above. In addition, prac- 
tically all torpedoes have the center of gravity below 
the center of buoyancy so that, if the torpedo rolls a 
little about its longitudinal axis, a restoring moment 
is produced. Then, if no torcpie is set up by an in- 
equality in rudder or fin area, reducing the problem 
of a circling torpedo to a static problem we may see 
that a heel will be produced in the turn and obtain 
the approximate magnitude of the heel. Consider the 
torpedo going around in a circle of radius R and 
heeled through an angle 0. Then, taking moments 
about the center of buoyancy of the torpedo, we find 
that, if the metacentric height is G, 

^ cos 0 = MgG sin 0 , 

where the left side represents the torque of the cen- 
trifugal force and the right side the restoring moment 
of the metacentric height. It then follows that the 
heel in the turn is given by 

4> = tan- (F) . (21) 


The condition of a dynamically stable torpedo 
going around in a steady turn may be ol)tained simply 
by setting a' = 9' = 0, which corresponds to a cir- 


This angle is not particularly small. If we consider 
the Mark 13 torpedo without a shroud ring, R ^ 250 
ft, I" = 50 ft per sec, so that 0 = 21.3°. 




120 


UNDERWATER RUN 


Remembering that for most torpedoes the running 
elevator position is down, when the torpedo is heeled 
over in a turn it is clear that the elevators will tend 
to have an opposite effect from the rudders and so 
tend to make a larger turning circle. Since the ele- 
vators are generally much larger than the rudders, 
this effect is of considerable importance. 

In order to calculate the effect of a steady heel we 
shall assume that the heel only affects the motion 
through the components of the elevator and rudder 
in the horizontal and vertical plane and that to this 
first approximation the hydrodynamic constants in 
the equations of motion are unaltered. This assump- 
tion is discussed in more detail in Section 6.5.2. From 
that section we see that the right sides of the equa- 
tions of motion are altered. From equations (56) and 
(57) in that section we see that the relation for as 
given by equation (18) above, becomes 



Cih — C,nW 


C ueCl-\-C\m 


sec</) — 


ClC ^e-\-Cr,iC\e ^ ^ ^ 

8o tan (/) , (22) 

^ ^ I \ / 


where 6o is the rudder angle in the turn and C^e, C\e, 
are the elevator coefficients. 

Thus, when the torpedo is heeled over in a turn, 
the running elevator angle is less down by the amount 
noted in the second term. From the equations of mo- 
tion in Section 6.5.2 we find that in a steady turn: 



(ClC fx-\-C\Cm)8Q COS0 — {ClCne-\-C\eCm)^r sill (f) 

CiCk - mCm ’ 


where is the mean elevator angle in the turn and 
may be recorded by some instrument or given by 
equation (22) above. It is seen, as was reasoned 
earlier, that since generally > 0, a roll angle in the 
turn increases the radius of the turning circle. Since </> 
varies with V~/R, will be a function of the velocity. 
This equation is the one referred to earlier relating 
Ck to C p. 

One method of obtaining the damping moment co- 
efficient Ck and damping force coefficient Cf(w = 

— (7/^) in the past has been from an analysis of turn- 
ing circle data. AVith a knowledge of R and assuming 
a relation between Ck and C f generally the ratio is ap- 
proximately Ck/Cf = 0.5; corresponding to the 
assumption that most of the damping force arises at 
the torpedo tail these constants were estimated. 
However, the effect of heel in the turn must be con- 
sidered since if it is neglected an artificially large Ck 
is deduced (since a heel and increasing Ck both in- 
crease the turning circle radius). This method will 
probably also be used in the future and it is therefore 
necessary to note the large effect of heel. 



PART III 


CONTROL SYSTEM 



Chapter 8 

GENERAL DISCUSSION OE CONTROLS 


8.1 PURPOSE OF CONTROLS 

I T WAS SHOWN in Section 7.2 that the Alark 13-2 
torpedo is dynamically stable with respect to mo- 
tion in the horizontal or steering plane. While this is a 
desirable state of affairs, it is not sufficient to guaran- 
tee good course-keeping; for any asymmetry in the 
torpedo or its propulsive plant, or any external dis- 
turbance will cause the torpedo to depart from its 
original course. It is necessary, therefore, to provide 
a correction which tends to restore the torpedo to its 
original course; this is supplied by a gyroscope and 
steering engine which operate the vertical rudders. 

The situation with regard to motion in the vertical 
plane (depth-keeping) is more critical since it is 
usually desired that the torpedo run within narrow 
limits of depth and quite close to the surface. Because 
of the inhomogeneous terms in equations (10) of 
Chapter 7, that is, because of the necessity for bal- 
ancing both the unbuoyed weight W — B and the 
moment Ba of the torpedo, a horizontal rectilinear 
path (12 = b = /? = 0) will not be obtained unless the 
attack angle a and the horizontal rudder or elevator 
angle ^ have precisely the right values, as given by 
equations (17) and (18) of Chapter 7. This adjust- 
ment requires a depth-control system which is sensi- 
tive to depth and which must therefore contain at 
least a hydrostat and a depth engine for operating 
the elevators. It will be shown in Section 12.1, how- 
ever, that a control of this type makes the torpedo 
dynamically unstable. It is therefore necessary to add 
an element which indicates incipient deviations from 
running depth before they develop to such an extent 
that the hydrostat cannot handle them. Such an ele- 
ment can be provided by a pendulum arranged to 
indicate deviations from running orientation angle; 
other arrangements involving gyroscopes and depth 
differentiators are possible as well. 

Apart from the maintenance of course and depth 
during the steady run, the control system must on 
occasion provide predetermined course changes in 
both the horizontal and vertical planes. Thus angle 
shots, in which the torpedo is brought to a new course 
after launching, and recovery from the initial dive 
must be taken into consideration in designing the 
control. The first is relatively simple, since it is only 
necessary that the gyro pick-off be preset so that the 


torpedo runs at a fixed angle to the gyro axis. The 
second, however, requires that the depth control be 
constructed so that the large decelerations inherent in 
the initial dive and recovery do not prevent it from 
bringing the torpedo to running depth quickly and 
without excessive broaching. 

8.2 TYPES OF CONTROLS 

There are two extreme control types which can be 
applied to either the course-keeping or depth-keeping 
problem. These are the linear or proportional system, 
and the limited or two-position system. A propor- 
tional control is one in which the rudder displacement 
is some linear combination of the magnitudes of the 
factors to be controlled (orientation, depth, etc.) and 
possibly their time derivatives. A two-position con- 
trol is one in which the rudder is always in one of two 
positions or in relatively rapid transit between them, 
the position selected again depending on the factors 
to be controlled. 

The torpedo together with a proportional control 
can be represented, at least for small deviations from 
running conditions, by a set of simultaneous linear 
differential equations with constant coefficients. The 
control is stable if a small disturbance produces a 
motion which damps out in time, that is if the real 
parts of all the characteristic exponents of the differ- 
ential equations are negative. If the control is not 
stable the motion will build up until the rudder mo- 
tion is limited by stops. When the instability is of an 
oscillatory character so that the exponents with 
positive real parts are complex, the rudder stops act 
as a limiter, and the control has many of the proper- 
ties of the two-position type. Such transitional cases 
are best considered along with the ideal two-position 
control. 

The two-position control operates stably when the 
rudder oscillates periodically between its two posi- 
tions and the torpedo motion is correspondingly 
periodic; then the motion following a small disturb- 
ance tends to resume this periodic character. It is 
possible for a two-position control to be unstable in 
the sense that the oscillatory motion of the torpedo 
increases in period and amplitude; this can, of course, 
also occur for an unstable proportional control which 
is operating in the transitional range. 


123 


124 


GENERAL DISCUSSION OF CONTROLS 


8.3 METHODS OF ANALYSIS: PROPOR- 
TIONAL CONTROL 


The most straightforward method of analyzing a 
proportional control consists in writing down the 
simultaneous linear differential equations that de- 
scribe the torpedo motion and the control system 
and solving them by the method of characteristic 
exponents already applied in Section 7.2. The tor- 
pedo equations have already been given in (10) of 
Chapter 7 and a relation between the rudder angle ^ 
and the torpedo parameters must be added. The 
steady-state values of the parameters can be found 
first and will in general be different from zero if in- 
homogeneous terms like w and h are present in the 
equations. The transient part of the motion can then 
be found as a sum of exponential terms like or 
depending on whether the distance in lengths s or 
the time t is used as the independent variable. These 
are to be substituted into the homogeneous equations 
that result when the steady-state terms are sub- 
tracted out. The requirement that a solution exist 
for the coefficients of the exponentials gives rise to 
an algebraic equation in the exponents p. This ‘‘secu- 
lar” equation can be solved by trial and error using 
synthetic division, or in other ways, and the coeffi- 
cients of the exponentials evaluated in terms of the 
initial conditions. In this way the explicit motion of 
the torpedo following a transient disturbance or dur- 
ing the recovery from the initial dive can be found. 

For many purposes it is sufficient to know simply 
that the control is stable, without knowing even the 
values of the characteristic exponents. In this case 
the Hurwitz criterion‘s can be applied directly to the 
coefficients of the secular equation. This criterion 
states that the necessary and sufficient condition that 
the real parts of all the roots of the algebraic equation 


flop” + aip”~^ + a2p”~2 an-A]) + a,, = Q (1) 

be negative is that all the quantities 

«! Uo 

^ 0 j ^ 1 j 

Oi do 0 
az a2 «! 

^5 CL^ CLz 

be greater than zero. 

“ Differ enlialgleichungen der Physik, Frank and von Mises, 

Vol. I, p. 163. 


az 

ao 




ai 

Qo 0 

0 • • 

• 0 


az 

Uo 

Clo • • 

• 0 

(2) 

0 

0 0 

. . . 

dn 



Another less direct but often more convenient 
method for dealing with the problem of the response 
and stability of the control system is based on an 
analogy with a linear feed-back amplifier. A feed- 
back amplifier is one in which a part of the output of 
a conventional electronic amplifier is fed back to the 
input. Great improvement in frequency response and 
freedom from distortion can be obtained in this way 
by proper design, but care must be taken that the 
circuit does not oscillate by itself or possess too slowly 
damped transients. Information as to the behavior 
of the circuit in this respect can be obtained from a 
study of the overall response function of the amplifier 
and feed-back network. Suppose that the feed-back 
network is disconnected from the amplifier input and 
a sinusoidal signal voltage of unit amplitude is ap- 
plied to the input. The magnitude and phase lead of 
the voltage that appears at the terminals of the feed- 
back network, when it is terminated by the input 
impedance of the amplifier, may be plotted as a 
function of signal frequency in polar coordinates. If the 
resulting curve and its image in the polar axis enclose 
the point which is unit distance from the origin along 
the polar axis, the complete feed-back amplifier will 
be unstable and oscillate with an amplitude which is 
limited by the non-linearities of the amplifier circuit’ 
The circuit is stable when the unit point is not en- 
closed, but if the curve passes too close to it the cir- 
cuit will possess slowly damped transients. These 
results were first obtained by Nyquist.*" 

The torpedo and control system can be regarded 
as a feed-back amplifier. For example, motion of the 
rudder produces a motion of the torpedo as a whole 
(amplifier) which produces a motion of the control 
system (feed-back network) and in turn actuates the 


RUDDER 



ENGINE 


ORIENTATION, 
DEPTH, ETC. 


Figure 1. Nyciuist criterion applied to torpedo and 
control system. 


rudder. If the control is disconnected from the rud- 
der, an overall response function can be defined in 
terms of the steering or depth engine motion pro- 
duced by a sinusoidal rudder motion of given fre- 

‘’H. Nyquist in Bell System T echnicalJ ournnl , Vol. 11 , 1932, 

p. 126. 


.METHODS OF ANALYSIS: TWO POSITION CONTROL 


125 


quency. The use of the Xyqiiist criterion in this con- 
nection has the great advantage that the overall 
response function can be broken up into two parts: 
one for the torpedo and one for the control system. 
Thus the torpedo response function need be calcu- 
lated only once and can be used in connection with 
various control response functions. ^Moreover, it can 
be seen from an examination of the response function 
what changes must be made in the control to improve 
the stability. 

The overall response function, once it is known, 
can also be used to find the torpedo motion resulting 
from some periodic external disturbance, such as a 
wavy sea. A straightforward, but less simple, calcula- 
tion enables one to calculate the transient response to 
an aperiodic disturbance by expressing it as a Fourier 
integral. 

8.4 METHODS OF ANALYSIS: 

TWO-POSITION 

CONTROL 

The ideal two-position control contains a limiter 
which switches the rudder instantaneously from one 
extreme position to the other when the controlling 
signal reaches a specified value from one direction 
and reverses the process when it reaches the same 
value from the other direction. A straightforward but 
difficult method for analyzing such a system consists 
in integrating the supposedly linear equations of mo- 
tion of the torpedo in intervals during each of which 
the rudder position has one of its extreme positions 
and matching boundary conditions at the instants 
when the rudder makes its traversals. 

A far simpler method is based on the fact that the 
torpedo response to rudder motion has the general 
properties of a low-pass amplifier so that high- 
frecjiiency components of the rudder motion are 
strongly attenuated. One can then assume that the 
rudder has a periodic square-wave motion, and this 
can be analyzed in a Fourier series. Because of the 
decrease in amplitude of the harmonics with fre- 
quency and the high-frecpiency attenuation, only the 
fundamental component of this series will be signifi- 
cant in the torpedo motion, which will therefore be 
very nearly sinusoidal. The response of the control 
and hence the phase of the limiter action can be pre- 
dicted from this motion, and this must of course agree 
with the originally assumed phase of the rudder mo- 
tion. Since the phase lags of the torpedo-response 
function and the control are generally increasing 


functions of the frequency, there will, in general, be 
one frequency for which the phases of the assumed 
rudder motion and the resulting limiter action agree ; 
this is the frequency at which the system oscillates. 

When the overall phase lag is an increasing func- 
tion of the frequency, the oscillation obtained in this 
wav is stable. This can be seen from the following 
argument. Suppose then an external disturbance mo- 
mentarily retards the rudder motion. This introduces 
lower frequency components into the rudder and 
hence into the torpedo motion. This in turn decreases 
the overall phase lag, making the limiter action take 
place more quickly and restoring the motion to its 
original periodic character. Similarly, a momentary 
speed-up of rudder action introduces higher fre- 
quency components, increases the phase lag and 
hence slows the motion down towards the original 
frequency. By the same argument, an oscillation fre- 



Figure 2. Effect of a low-frequency external dis- 
turbance on a stably oscillating two-position control. 

quency which occurs when the overall phase lag is a 
decreasing function of frequency is not stable. When 
this occurs, the phase lag will generally increase 
eventually with frequency and produce a higher fre- 
quency oscillation which is stable. (3n the other hand, 
if an external disturbance causes the frequency to 
start decreasing from the unstable point, it is possible 
that an unstable oscillatory motion will result in 
which the period and amplitude increase with time. 

It is possible to find in a simple way the response of 
a stably oscillating two-position control to an exter- 
nal disturbance which is of low frequency compared 
to the control-oscillation frequency and whose ampli- 
tude at the limiter is small compared to the control- 
signal amplitude. Suppose the control-signal ampli- 
tude at the limiter is S and the disturbance is a 
steady signal of magnitude s at the limiter. If the 
limiter action takes place when the total signal 
crosses the zero value, as is usually the case, the effect 


126 


GENERAL DISCUSSION OF CONTROLS 


of the disturbance is to cause the rudder to spend 
slightly more time in one position than the other; 
this adds a small component of rudder in that direc- 
tion without affecting the phase of the limiter or 
rudder action, and hence without affecting the fre- 
quency of the steady oscillation. The rudder then 
spends a fraction + s/wS) of a cycle in one direc- 
tion and a fraction — s/ttS) of a cycle in the other 
direction, so that the net steady rudder component 
is 2s^o/TrS, where is the rudder throw. But the 
amplitude of the fundamental component of the 
rudder motion is 4^o/7r so that the ratio of steady to 
fundamental component of the rudder is s/2S. 

While the proof given here applies only to ex- 
tremely slow disturbance frequencies, it has been 
shown that this applies for all lower frequencies and 
for amplitude ratios at the limiter of the order of 3^ or 
less. Thus the ideal two-position control acts like a 
linear amplifier for external disturbances, and the 
oscillations can be ignored from this point of view. 
This approach is useful in considering response to 
waves and the effect of initial conditions. Also it is 
this approach which permits the treatment of on-off 
systems subject to disturbances as though they are 
proportional systems. This enables one to calculate 
practically all the behavior characteristics of a tor- 
pedo with an on-off system that one could calculate 
for a torpedo with a proportional system. 

As a simple but instructive application of this very 
useful theorem we shall calculate the trajectory in 
the horizontal plane of a dynamically stable torpedo 
with proportional depth controls and two position 
(on-off) steering when the torpedo is heeled over 
clockwise by an angle </>. Essentially, we shall look for 
the final course angle of the torpedo. 

Clearly, for such a torpedo we have the case of an 
on-off system subject to a disturbance (caused by 
the steady component of the elevators in the hori- 
zontal plane) which is of lower frequency than that 
of the on-off system, in fact in this case the distur- 
bance is of zero frequency. 

One can then readily write down the equations of 
motion in the horizontal plane (where the hydro- 
dynamic constants are taken to be independent of 
heel in this approximation). 

TTZ 12 + W7 2 a' + Cia = — Cx8 cos 0 Cxe^r sin 0 , 
nl2' + C^12 + Cma = C^8 cos 0 — C^e^r sin 0 , 

8 = —7/3. 


8 is the steady rudder component, is the steady 
running elevator angle. From the previous theorem 
it is evident that 

25o 

7r/3o 

where 5o is the rudder throw and /3o is the amplitude 
of the yawing oscillations. The general solution of 
these differential equations is 

a = a -{- 0:26^^® + a^e^^ 0:4 , 

^ + /32eP^® + . 

One may take as the initial conditions of the motion 

/3i = = ai = 0. 

If the body is dynamically stable the real parts of 
Pi, P2, and pz must be negative. 

For example, for the Mark 13 torpedo with shroud 
ring the real parts of the characteristic exponents 
have the magnitudes —2.2, —0.51, —0.51, the last 
two being the same since two of the roots are complex 
conjugates. 

Since these exponents are to be multiplied by s, it 
is clear that the corresponding terms will damp out. 
Since we are interested in what will happen for large 
values of s (the course deviation at the end of many 
torpedo lengths), it is evident that (for 5 large) 

q; = 0:4 , 

(3^aA, 

since and ^4 are constants independent of s. 

Substituting into the equations of motion, one finds 

^ _ {CxeC^, — C^,eCx)^rSm(j) 

(C,C„ + CxC„,) ’ 

Q _ (C^ufC/ + (7xf(7w)^r tan 0 

' liCiC, + CxC„d 

Since the center of pressure of the rudder and ele- 
vator lift forces is almost exactly the same distance 
from the center of gravity 

^ = X 

Cx Cxe 

where X is the fraction of a torpedo length by which 
the center of pressure of the lift forces of the elevators 


INTERACTION OF DEPTH-KEEPING, STEERING, AND ROLL 


127 


and rudders (approximately the elevator or rudder 
stock) lies aft of the center of gravity. 

With this consideration one finds 



1^4 = 


(r'm H“ tan 0 C\, 


+ \Cl)C\ 




tan 0 . 


Remembering that y 


2 5o 
TT jSo ’ 


TT C^\e 
2 ^ 


jSo tan 0 . 


Xaturally for heel angles of interest (which are gener- 
ally small) one can replace tan 0 by 0 in radians. Thus 
the torpedo travels with zero yaw angle and with the 
average angle between the torpedo axis and the di- 
rection of the gyroscope setting being (3i. The trajec- 
tory angle is given by 6 = a -\- ^ = ( 34 . Thus the 
angle the course pursued makes with the set direc- 
tion, when the torpedo is heeled over, is given by (Si. 

The course deflection or the distance between the 
actual torpedo position and its desired position is 
given by 



— /3oS tan 0 . 
5o 


For the Mark 13 torpedo with shroud ring, using /3o 
as given in Chapter 10, /3o = 0.77°, C\e/C\ = 10, 
= 2.2°, 6o = 9°, we find 

6 = (34 = 3° tan 0 


primarily to lack of complete torque balance of the 
propellers), it is seen that this is a very large effect 
compared to the restrictions commonly imposed on 
torpedo gyroscopes. 

W e see from this result that a clockwise heel will 
produce a nose-right course deflection for down ele- 
vator running angles. 

It is also seen that decreasing (the elevator run- 
ning angle) and decreasing the time lag in steering 
(which from Chapter 10 is seen to diminish (3o) will 
diminish the deflection due to heel. 

The reason for the torpedo to run at an angle ^4 
between its axis and the direction of the gyroscope is 
that this angle is necessary to produce the steady 
component of the vertical rudder, since it behaves 
like a proportional system and the steady component 
is proportional to /3, which will overcome the steady 
effect of the elevators in the horizontal plane. 

The discussion thus far has been restricted to ideal 
two-position controls. An actual control may differ 
from this in three principal ways. First, the rudder 
does not switch instantaneously, but requires a finite 
time to travel from one position to the other. Second, 
there may be a time delay in the transmission of the 
switching signal from the limiter to the rudder. Third, 
the control values at which the switching action takes 
place need not be the same for the two directions. All 
three of these effects introduce additional phase lags. 

The Fourier analysis method outlined above may 
also be applied to the study of unstable proportional 
controls in which the amplitude of the rudder oscilla- 
tion is limited by stops. In such a transitional sys- 
tem, the amplitude and phase relations between 
rudder and torpedo motion must usually be consid- 
ered simultaneously. 


or the deflection is given by y = (3 tan 0/57.3)s tor- 
pedo lengths. One may also say that the deflection is 
given by 

Deflection = 52.7 tan 0 mils 

= 52.7 0mils (for small 0 where 0 is in 

radians) . 

Thus, if the Mark 13-G torpedo is heeled over about 
5.5° in its steady run, it will suffer a course deflection 
of about 5 mils. 

This result is in good agreement with verbal re- 
ports of statistical studies made of runs at the New- 
port Torpedo Station. 

Since most torpedoes have some heel (probably due 


8.5 INTERACTION BETWEEN DEPTH- 
KEEPING, STEERING, AND ROLL 

It is assumed throughout most of this report that 
the motions of the torpedo in the vertical and hori- 
zontal planes can be treated separately. It is evident, 
however, that if the torpedo rolls or heels, the vertical 
rudders act to a certain extent as elevators (horizon- 
tal rudders), and vice versa. The resulting interaction 
between depth-keeping, steering, and roll may be of 
importance in two situations. First, in the recovery 
from the initial dive, large rolls are encountered 
which may affect the operation of the depth controls. 

Second, in steering with a two-position control, a 
certain amount of roll having the same frequency as 


128 


GENERAL DISCUSSION OF CONTROLS 


the steering oscillation is produced by the vertical 
rudder action and the displacement of the center of 
buoyancy above the center of gravity of the torpedo. 
Attempts are made to minimize this roll by making 
the upper vertical rudder larger than the lower one 
and the lower vertical fin larger than the upper one. 
However, the elevator effect of this vertical rudder 
motion is roughly proportional to the product of the 
roll angle and the rudder displacement and hence has 
twice the frequency of the steering oscillation. Thus 
in the absence of heel the steering action can influence 


the depth-keeping, but this influence will not react 
back significantly on the steering since it has twice 
the frequency. When the torpedo is heeled over, how- 
ever, there is also an elevator effect component of the 
vertical rudder motion that has the same frequency, 
and in this case one should solve the combined equa- 
tions of motion. Although little quantitative work 
has been done on this effect, it seems likely that it 
provides the explanation for the synchronous depth 
and roll oscillations sometimes observed when the 
torpedo is heeled over. 


Chapter 9 


PROPORTIONAL CONTROL 


T his chapter applies the methods discussed in 
Section 8.3, to the detailed consideration of the 
horizontal steering of the i\Iark 13-2 torpedo. It is 
assumed that the torpedo is provided with a linear 
steering control in which the steady-state deflection 
of the vertical rudder is proportional to the deviation 
of the gyroscope axis from the desired heading. This 
will not be true in general for varying gyro signals 
since some delay in transmission is inevitable with 
rapid variations. Thus, in addition to the torpedo 
equations (10) of Chapter 7, with w = h = 0, there is 
a connection between the orientation angle 6 and the 
rudder angle ^ (both in radians) which may be writ- 
ten as follows: 


ri -h ^ = —7/3 . (1) 

This equation would result, for example, if a pneu- 
matic transmission possessing viscous friction and 
stiffness described by a time constant r were used 
to transmit the gyro signal to the rudder. The equa- 
tions of motion of the system ma}" then be written 

7712^' + Ga -j- 777 = —Cx^ , 

CmOi + nf2' Ca'^I = , (2) 

= -7/3, 

where a = Vr/L 


9.1 HURWITZ CRITERION 

In this method a substitution of the form is 
made for each of the variables, resulting in the fol- 
lowing secular determinantal equation : 


m 2 P + Cl 771 Cx 0 

Cm np + Ck ~Cfj 0 

0 0 o-p -f- 1 7 

0 1 0 -p 


= 0. (3) 


This may be rewritten as a fourth degree algebraic 
equation in the characteristic exponent p : 


Gcp'* + cqp^ + a2P“ + ttaP + 04 = 0 , 


Qo = 777 2 77 a, 

ai = 777 271 -f <j(7lCi + 777 2Ck) , 

«2 = { 71 C 1 + 777 2Ck) + criCiCx ~ 777Cm) , 

O.Z = CiCk ~ 777Cm + 7772C , 

04 = 7(GG + GnCx) . (4) 

Application of the Hurwitz criterion (2) of Chapter 8 
to this equation is equivalent to the requirement that 
all of the coefficients be positive and that 


UiU2a3 — OoUs” — ar«4 > 0 . (5) 

For a dynamically stable torpedo such as the i\Iark 
13-2, equation (16) of Chapter 7 states that CiCk — 
77iCm > 0; thus in this case all of the coefficients are 
positive, and equation ( 0 ) is by itself a necessary and 
sufficient condition for stability of the controlled 
torpedo. 

Some of the numerical parameters representative 
of the Alark 13-2 torpedo have been given in Section 
7.1, and Figure 1 of Chapter 7. The inertia parameters 
777 i, 7712, and 71 may be calculated from the mass and 
moment of inertia of the torpedo, with the correc- 
tions for entrained water being made approximately 
on the basis of Lamb’s calculation for prolate sphe- 
roidal shapes. Provisional numerical values are col- 
lected below for this torpedo with plain and shroud 
ring tails. These are intended for use as illustrative 
parameters and are not definitive magnitudes. 



Plain 

Shroud 

mi 

1.80 

1.84 

m 2 

3.11 

3.20 

n 

0.176 

0.188 

Cm 

0.6.50 

0.279 

Cm 

0.295 

0. 173 (horizontal 
elevator) 

Cm 

0.0.30 

0.017 (vertical 
rudder) 

Cl 

2.01 

2.30 

Cx 

0.606 

0.344 (horizontal 
elevator) 

Cx 

0.061 

0 . 034 (vertical 
rudder) 

C K 

0.44 

0.48 

Cf 

1.01 

1.16 

m 

0.79 

0.68 




129 


130 


PROPORTIONAL CONTROL 



Substitution of these numbers into equations (4) 
and (5) shows that the motion is stable for any value 
of 7 if there is no time delay {a = 0). For cr = 0.5, 
which corresponds to r = 0.10 sec for 40 knots and 
T = 0.12 sec for 33 knots, the plain tail torpedo is 
stable if 7 < 28, and the shroud tail torpedo is stable 
if 7 < 99. Thus the improvement in dynamic sta- 
bility of the uncontrolled torpedo produced by the 
addition of the shroud ring (see end of Chapter 7) is 
reflected here as an increase in the range of control 
parameters over which stable operation occurs. 

92 NYQUIST CRITERION 

In this method the 6 response to a sinusoidal rudder 
motion ^ is calculated first, and then the rudder mo- 
tion produced by this 6 motion through the con- 
trol. The two together give the overall response of 
the controlled torpedo, and from a plot of this as a 


function of frequency the stability may be inferred. 
In performing the calculation it is convenient to use 
complex exponentials rather than sines and cosines. 
Thus if ^ is substituted on the right 

sides of the first two of equations (2) and the steady 
state solutions Q = and d = found, the 

complex ratio for example, may be written as 

I where the magnitude is the amplitude ratio 

and xp is the phase lead in radians of over The (3 
response is readily obtained from the fourth of equa- 
tions (2), and do may be found from the third of 
these equations. The complex overall response func- 
tion is then given by the product of the three factors: 



From Section 7.2, the relation between the fre- 
quency V in cycles per sec, the angular frequency co in 


NVOUIST CRITERION 


131 



Figure 3. Logio magnitude /io/^o versus k. 



0.! .2 .3 .4 .5 .6 .7 .8 .9 I 2 3 4 5 6 7 8 9 10 

K 


Figure 4. Phase of h/^o versus k. 


L06,o MAGNITUDE FOR VERTICAL RUDDERS 


132 


PROPORTIONAL CONTROL 



bJ 

Vi 

< 

X 

Q. 


Figure 5. Phase of and logio magnitude of for shroud-tail steering versus k. 


radians per sec, and the dimensionless quantity k is 

‘lirv = oi = — k . 

I 

Thus, for the Mark 13-2 torpedo at 40 knots, v = 
0.79A', and at 333^ knots, v = 0.66A-. 

The solutions of the first two of equations (2) are 


ao _ (TnCfi + CkC\ -f- iknC\) 

^0 A ’ 

_ {CmC\ T CiCfjt + ikm^Cf^ 


A = nm o/:- — {CiCk — mCm) — ik{nCi-\-m2CK) • 


The last two of eciuations (2) give 

^ _ I 

9.0 ik ’ 



= - T 
/^o IT ika 



The quantity /3o,To = (9o/^o)i(3o/i2o) can be computed 
from (7) and (8) with the numerical values of Section 


9.1; its phase and the logarithm of its magnitude are 
plotted as a function of k for the plain and shroud 
ring tails in Figures 1 and 2. Similar curves of ho/^o, 
where h is the lateral deviation from straight course, 
are plotted in Figures 3 and 4 (see Chapter 12). The 
left ordinate scales for Figures 1 and 3 refer to the 
horizontal elevators and the right ordinate scales to 
the vertical rudders. 

Before completing the study of this control by 
NyquisFs method, it is necessary to discuss a pe- 
culiarity of the torpedo problem which does not arise 
in most other control problems. This is the singu- 
larity in do/^o at k = 0, which makes it impossible to 
apply Nyquist’s criterion directly. Physically, this 
singularity means that if the rudder is oscillated 
slowly enough, the angular excursions of the tor- 
pedo can be made arbitrarily large with fixed rudder 
amplitude; this is to be expected so long as the non- 
linearity of the equations for such large angles is 
neglected. It can be shown that the way in which to 
deal with this singularity is to bring the phase of the 
/3o/^o response function from the smallest k of prac- 
tical interest to k = 0 along the dotted curves indi- 
cated in Figure 2 and keep the magnitude constant. 
This corresponds to avoiding the singularity by re- 
placing by for small k, where the contour in the 
complex p-plane is taken along the imaginary axis 


NYQUIST CRITERION 


133 



Figure 6. Phase of and logio magnitude of ^o*/^o for plain tail steering versus k. 


(p = ik) in to a small value of k, and then around the 
origin to the right. When this procedure is followed, 
Xyquist’s criterion may be used."^ 

The phase and the logarithm of the magnitude of 
the overall response function (6) are plotted as a 
function of k in Figures 5 and 6, the circuit of the 
origin being indicated by dotted curves. The control 
parameters chosen are u = 0.5, y = 60, for which, 
according to Section 9.1, the shroud tail torpedo 
should be stable and the plain tail not. The inserts of 
Figures 5 and 6 show schematically (not to scale) the 
shape of the Xyquist diagrams in polar coordinates, 
the positive real polar axis being to the right and the 
unit point being marked with a circle; it is evident 
that these diagrams agree with the results of Section 
9.1. A comparison of the curves and the inserts shows 
that it is necessary to examine the response function 
only in the region of k near which the magnitude is 
unity and the phase zero. If the phase crosses zero 
(from positive to negative as k increases) at a smaller 
value of k than that at which the magnitude crosses 
unity (decreasing as k increases) so that the magni- 
tude is greater than unity when the phase is zero, the 
system is unstable; otherwise, it is stable. This last 

^ It is worth noting, as is easily proved, that Nyquist’s cri- 
terion gives incorrect results when applied to torpedoes that 
are dynamically unstable; this does not, however, affect its 
application to the Mark 13-2 torpedo. 


statement of the Xyquist criterion is of course special 
to this problem, but it does apply to a large group of 
torpedoes and controls; in case of doubt the whole 
of the X’yquist diagram should be drawn. 

It might appear from the foregoing discussion that 
the Hurwitz criterion is easier to apply than the 
Xyquist criterion. This is generally true if only a 
single control and a single type of motion need be 
considered. But if, for example, one wishes to con- 
sider both proportional and two-position systems or 
if one wishes to compare a number of different types 
of proportional control on a single torpedo, then the 
decomposition of the overall response function into 
a product of torpedo and control response functions 
saves a great deal of numerical work. It should also 
be noted that the relatively simple fourth degree 
equation (4) for p becomes a sixth or higher degree 
equation for a pendulum type depth control. 

Another advantage of the Xyquist method is that 
it is possible to do some problems that cannot be 
treated at all by the Hurwitz method. These are 
problems in which frequency-independent time de- 
lays occur or in which part of the overall response 
function, say that of the steering or depth engine, is 
measured empirically. Although the Xyquist method 
has been rigorously justified only for systems that 
can be described by a set of linear differential equa- 
tions of finite order, it seems very plausible that the 


134 


PROPORTIONAL CONTROL 


method is applicable to systems with time delays or 
mild nonlinearities. As an example, the friction- 
stiffness time constant assumed in the problem 
worked out above could be replaced by a fixed time 
delay r simply by introducing an additional phase 
lag 21: V 7 = k<j radians into the overall response 
function. This could not, on the other hand, be 
treated by the Hurwitz method since a fixed delay 
can only be represented by a differential equation of 
infinite order. Thus, in order to represent 

^(s + 0-) = —yO{s) 


in place of the third of equations ( 2 ), one would re- 
quire the differential equation: 

, , ,, , 0 - 2 ^' , , 

C + or? ••' = —yd . 

iU 1 

The convergence of this equation would have to be 
investigated in particular cases of interest. It should 
be noted, however, that in order to calculate the mo- 
tion and trajectory of the torpedo the character- 
istic exponents must be determined. 


Chapter 10 

TWO-POSITION CONTROL 


T his chapter applies the second (approximate) 
method discussed in Section 8.4 to the detailed 
consideration of the horizontal steering of the Mark 
13-2 torpedo. It is assumed that the torpedo is pro- 
vided with a two-position steering control Avhich 
possesses characteristics similar to those of the stand- 
ard pallet mechanism. Thus, if the desired heading is 
^ = 0° and the gyroscope axis is in this direction, the 
mechanism is set, into motion when the heading be- 
comes rtiSc, and the rudder moves very quickly from 
stop to stop a time r after this. 



Figure 1. Phase lags caused by torpedo motion, the 
finite actuation angle dc, and the time delay. 


taken into account by increasing the time delay by 
half the transit time and correcting the amplitude of 
the fundamental component of the rudder motion in 
accordance Avith the departure from square-AvaA^e 
form; the second of these effects is usuall}" negligible 
Avhereas the first is not. 

The relationships betAveen the phase lags due to 
the torpedo motion, to the finite actuation angle /3c, 
and to the time delay r are shoAvn schematically in 
Figure 1. It is clear from this that the sum of the 
three phase lags must be 180° in order for a steady 
oscillation of this tA^pe to persist. The torpedo phase 
lag is the negatiA^e of the phase of j3o/^o plotted in 
Figure 2 of Chapter 9, and the phase lag due to the 
time delay is simply 27rrr = ka radians, Avhere again 
a = Vt L The actuation phase lag is seen from 
Figure I to be sin“^ (^c/ M, where (So is the amplitude of 
the oscillation of the torpedo heading. (So may be com- 
puted from Figure 1 of Chapter 9 (using the right 
ordinate scale) for any k value b}" multiplying the 
magnitude of (So/'^o by d^o/V, AA’hich is the amplitude 
of the fundamental component of the rudder motion. 

The rudder throAv in the Mark 13-2 torpedo is 
^0 = 12.1°. Figure 2 shoAvs the three component phase 
lags and their sum as functions of k for the shroud 
tail torpedo, under the assumption that /3c = 0.5° and 
(7 = 0.5, AAdiich corresponds to r = 0.10 sec at 40 
knots and r = 0.12 sec at 333^ knots. Figure 3 sIioaa’s 
similar curves for the plain tail torpedo. Since the 
total phase lags increase AA'ith v or k in each case, the 
180° crossings represent stable oscillations (see Sec- 
tion 8.4). These crossings happen to occur at k = 0.76 
for both tails. A similar calculation for a = 0.4 cor- 
responding to T = 0.10 sec at 333^ knots, gh^es cross- 
ings Avhich were used to construct the folloAving table. 

Mark 13-2 Torpedo 


A square-Avave rudder motion of amplitude and 
of definite frequency or k value (see Section 9.2) is 
assumed, and the torpedo motion corresponding to 
its fundamental component is calculated. The phase 
at Avhich the (S motion causes the control to operate 
the rudders is found, and this corrected for the time 
delay r must agree with the originally assumed phase 
of the rudder motion. If the time of transit of the 
rudder from stop to stop is appreciable, it may be 


Period (7’) in seconds and amjilitude (do) for steering. 
^0 = 12.1°, dc = 0.5°, T = 0.1 sec. 


40 knots; 
333^ knots: 


Shroud tail 
T do 

1.66 0.82° 

1.89 0.77° 


Plain tail 
T do 

1.66 1.98° 

1.84 1 . 73° 


Both the assumed parameters and the calculated 
periods are in good agreement Avith obsei’Amtion; no 
experimental data are available on /3o. 


135 


136 


TWO POSITION CONTROL 



Figure 2. Three component phase lags and their sum as functions of k for the shroud tail torpedo. 



0.1 .2 .3 .5 .6 .7 .8 .9 I 2 3 4 5 6 7 8 9 10 


K 

Figure 3. Three component phase lags and their sum as functions of k for the plain tail torpedo. 



TWO POSITION CONTROL 


137 


It will be noticed that increase in dc and decrease in 
^0 increases the actuation lag, increase in r or a in- 
creases the time lag, and removal of the shroud ring 
increases the torpedo lag. All of these changes increase 
the overall phase lag and hence increase the period of 
oscillation. Thus the close agreement between the 
values for the periods with and without the shroud 
tail is due to a compensation between the increase in 


torpedo phase lag caused by removal of the shroud 
ring and the decrease in actuation phase lag caused 
by the increase in /So that accompanies this removal. 
In similar fashion, increase in speed causes k to de- 
crease for given v and hence causes the torpedo lag 
and the actuation lag to decrease; the time lag is un- 
changed. Thus the overall lag and hence the period of 
oscillation decrease. 


STEERING CONTROL 


T he steering of the Mark 13-2 torpedo has been 
used to provide a numerical example of a pro- 
portional control in Chapter 9 and of a two-position 
control in Chapter 10. The methods available for 
constructing these controls will be discussed in Sec- 
tion 11.1; Sections 11.2 and 11.3 deal with problems 
that arise in maneuvering in the horizontal plane. 
Before proceeding further, it is worth emphasizing 
that the use of a gyroscope as a standard of course 
direction is fundamental to all devices intended to 
steer the torpedo or to maneuver it in a predeter- 
mined manner. The long time errors that make an 
uncorrected gyro unsuitable for use as a compass are 
of little importance in the case of a torpedo, where 
the short duration of the run makes it possible to use 
a gyro of quite moderate dimensions and speed. 

11 1 AVAILABLE METHODS 

The problem to be discussed here is that of making- 
available a force large enough to move the vertical 
rudders in accordance with the information supplied 
by the gyroscope, without producing a reaction on 
the gyro large enough to cause it to process appre- 
ciably. 

The standard pallet mechanism puts the valve of a 
pneumatically operated steering engine in one of two 
positions in accordance with the orientation of the 
outer gimbal of the gyro with respect to the torpedo 
body. The energy for moving the valve comes from 
the propeller shaft, which oscillates the pallet in such 
a way that it is tripped in one direction or the other 
])y a blade on the gimbal without reacting back ap- 
preciably on the gimbal. Since the pallet carries the 
information to the valve only once per cycle, there is 
an inherent time delay in this system which is of the 
order of the period of oscillation. In addition, the 
torpedo must deviate from course by a finite angle 
before the pallet is tripped, and machining tolerances 
prevent this angle from being as small as might be 
desired. As was shown in Chapter 10, both of these 
effects lengthen the steering period, and hence in- 
crease the amplitude of horizontal oscillation and of 
roll. 

Another device that operates in the same general 
manner has recently been developed for use in some 
electric torpedoes by the Westinghouse Electric and 


Manufacturing Company. A light roller attached to 
the gyro gimbal is arranged so that it makes a con- 
tact whenever the torpedo is off course in one direc- 
tion, and breaks it when it is off course in the other 
direction. The contact is connected in series with a 
relay which passes current through one or the other 
of two solenoids which move the rudders in opposite 
directions. In this case the actuation phase lag (see 
Chapter 10) is negligible, but the time delay can still 
be appreciable if the relay and solenoids are not 
properly designed. 

A pneumatic control that has a more rapid re- 
sponse than the pallet mechanism and can also be 
used as a proportional control has been developed by 
Columbia University and the American Can Com- 
pany for use on the JMark 25 torpedo. This consists 
of a semicircular blade mounted on the gyro outer 
gimbal and coaxial with it, which interrupts two jets 
of high-pressure air moving radially outward at op- 
posite ends of the diameter of the semicircle. Air is 
supplied from a single source to both jets and, after 
traversing the small gap in which the blade moves, 
goes into two receptive orifices which lead it to oppo- 
site sides of the steering engine piston. When the gyro 
is centered with respect to the torpedo axis, the blade 
interrupts both air streams equally, and the pressures 
developed on the two sides of the piston are equal. A 
small motion of the blade causes more air to go to 
one side of the piston than the other and moves the 
piston and hence the vertical rudders accordingly. 
The quite small reaction of the air stream back on 
one edge of the blade is nearly cancelled by that on 
the other edge. 

The hydrodynamic pressure on the vertical rudders 
produces a restoring force which is approximately 
proportional to the rudder deflection. Since the piston 
force is also nearly proportional to the blade deflec- 
tion, the entire system acts as a proportional control 
with properties somewhat like those assumed in 
Chapter 9; the time constant assumed there is sig- 
nificant if the connecting air lines from the blade to 
the engine are more than a few inches in length. In a 
well-designed practical system, with line pressure of 
the order of 500 psi and piston area of the order of 
one scpiare inch, the hydrodynamic restoring force is 
so small that for the time constant encountered the 
quantity 7 in equation (1) of Chapter 9 can be made 


MANEUVERING: APPROACH TO TURN 


139 


to exceed the maximum value at which the S3^stem 
operates as a stable proportional control (see Chapter 
9). It then operates as a transitional s\"stem and for 
large 7 maj" be treated as a two-position control. 

A proportional S3"stem ma3^ be developed along 
these lines b3' loading the steering engine output with 
springs large enough to reduce 7 to the stable range. 
This is referred to as a force-proportional control. It 
is the force exerted b3" the steering engine rather than 
its position which is proportional to the blade de- 
flection, and the engine or rudder position is propor- 
tional to this onh" if the restoring force has a fixed 
proportionalit3" constant and static or Coulomb fric- 
tion can be neglected. The h3"drod3mamic restoring- 
force depends on the speed but is generally much 
smaller than the spring restoring force, which can be 
made quite constant and reproducible. 

While in principle positioning type engines and 
phase advance schemes may be applied to the steer- 
ing problem, this has not been attempted thus far, 
and it seems unlikel3" that it will be in the near future. 
These techniques will be discussed in Section 12.1 in 
connection with depth controls. 


with rudder amidships. Since the Mark 13-2 torpedo, 
with or without the shroud ring, is dynamically stable 
(and all other torpedoes that have been studied ap- 
pear to be as well) , the methods of this section are of 
practical interest. If it were ever necessary to study a 
dynamically unstable torpedo, however, the present 
treatment would be inadequate, and it would be 
necessary to include the steering control as a funda- 
mental factor in the motion. 

The equations to be solved are (10) of Chapter 7 
with w = b = 0: 


llloOL C lOL -\~ lllQ. — — Cx^O J 

Cmd + • 



The solutions of these inhomogeneous equations are 
the same as for the homogeneous equations con- 
sidered in Section 7.2, except for the addition of con- 
stant terms : 


a = -|- 0:26^2* + 0:3 , 


o = + 1706^2® + 9.3 . 



11.2 MANEUVERING: 

APPROACH TO TURN 

It is sometimes useful to be able to predict the tra- 
jectoiy of a torpedo in the horizontal plane when the 
rudder is manipulated in accordance with some pre- 
determined plan. The two examples to be worked out 
in this section and the next illustrate the principal 
features of calculations of this type. In the first ex- 
ample, it is assumed that the torpedo is running on a 
straight course when the rudder is suddenly thrown 
hard over and maintained that wa3v The torpedo 
then makes the transition from the straight run to 
the steady turning circle. In the second example 
(Section 11.3), it is assumed that the torpedo is in its 
steady turn when the rudder is suddenly brought 
amidships, and the transition to a straight course is 
to be found. These examples provide the elements of 
the calculation of an angle shot, and extensions of the 
calculations to other problems of practical interest 
can be made. 

For illustrative purposes it is permissible to leave 
the actual steering mechanism out of consideration 
and to assume that the only rudder motion is a rapid 
transit from center position to hard over, or vice 
versa. This gives sensible results if the torpedo is 
dynamically stable (see Section 7.2) since then the 
undisturbed torpedo will run on a straight course 


The characteristic exponents pi and p2 are as given 
in equation (15) of Chapter 7. 

For the first example, in which the torpedo is 
going from straight course into a turn, ^0 is the full 
rudder angle, and the initial conditions (at s = 0) are 
^ = a = 9 = 0 . In this treatment the roll angle of 
the torpedo in a turn is neglected. The initial values 
of a' and 9 ' are also required in order to evaluate the 
coefficients in (2), and these may be obtained from 
the equations of motion (1) at s = 0; the results are 

/ Cx^o 

0^0 = — , 

m2 

(3) 

^ 

ILq — . 

n 

The values of 0:3 and 9.3 may be obtained by substi- 
tuting (2) into (1) and equating terms independent 
of s. The solution of the resulting equations is 

o _ _ (C/yCx -f- m Cfji)^o 

' ~ {CiCk - rnCm) ’ 

( 4 ) 

^ _ jCiCfi -j- CwCx)^o 

“ {C,Ck - mC„) ■ 


140 


STEERING CONTROL 


The initial values of a, 9. and their derivatives then 
give the following equations for the other coeffi- 
cients of (2) : 


+ «2 = — «3 , 


^^1 + ~ , 


Pio:i + 


7n2 ’ 


PlQl + P2f^2 


n 



The solutions of equations (5) are 

(~C 2 + ^ 2 ^ 3 ) 

ai = , 

(Pi - P 2 ) 

o ^ (C\^o/m2 - Vi(3z) 

(Pi - P 2 ) ’ (0) 

^ _ {C^^o/n + P2f^3) 
iPi - P 2 ) 

_ {C,i^o/n + 7 ) 1 ^ 23 ) 

ii2 • 

iPi - P2) 


Equations (2), (4), and (6) give the complete solu- 
tions for a and 9 as functions of the distance s along 
the trajectory measured in torpedo lengths. 


y 



coordinates along and perpendicular to the original 
course. 


Equations (7) may be integrated to give 


=/ 


x(s) = / cos 6{z)dz , 


■I. 


y{s) = sin d{z)dz . 



Since 6 will be given in terms of exponentials of its 
argument, the integrals in (8) cannot be evaluated 
analytically except as infinite series. For many pur- 
poses, the leading terms of these series are sufficient; 
these are obtained by putting cos 6 = 1, sin d = d, 
when 

a:(s) = s , y{s) = J" 6{z)dz . (9) 


The trajectory given by (9) is accurate so long as 
the course angle 6 does not become large. Thus, for 
the errors to be less than 10 per cent, 6 must be less 
than 0.45 radian = 26°. It is evident, however, that 
since the torpedo is going into a circle, 6 will become 
arbitrarily large after a sufficiently long time. This 
does not affect the usefulness of (9) so long as the 
motion becomes circular before 6 is large since then 
the steady turning circle can be fitted on to the tran- 
sient phase of the motion which is described by equa- 
tions (9). An explicit expression for the course angle 
is obtained by integrating 9 as given by (2) and 
adding a : 


d{s) = -J- 0 : 26 ^”^ -f- 0^3 “h 


(e 


+ 


Pi 

92 - 1 ) 

P2 


T 9^s . (10) 


The asymptotic form of this as s becomes large is 




+ ( 11 ) 

\ P\ P 2 / 


The orientation angle /3 may be obtained by inte- 
grating the equation (3' = 9, and the course or tra- 
jectory angle is then 6 = ^ a (see Figure 2 of 
Chapter 7). Now if x and y are rectangular coordi- 
nates along and perpendicular to the initial course of 
the torpedo, measured in lengths, the following rela- 
tions are valid: 

dx ^ (hj . ^ 

— = cos 6 , — = Sind 

ds ds 


which makes the trajectory a circle of radius (meas- 
ured in lengths) p = 1 / 93 . 

The question as to the accuracy of (9) may now be 
put as follows: does the d{s) given by (10) approach 
the asymptotic form (11) sufficiently closely before 
6 becomes large enough to invalidate the approxima- 
tions made in deriving (9)? Retaining the 10 per cent 
criterion of accuracy, the requirement is that 


( 7 ) 


S 0.1for»(s„) = 0.45. 


( 12 ) 


MANEUVERING: PULLOUT FROM TURN 


141 


The smaller exponent pi, given by the upper sign in 
equation (15) of Chapter 7, is used since it furnishes 
the more stringent criterion. In calculating so from 
the second part of (12), the asymptotic form (11) 
may be used. 

If now it is assumed that the criterion (12) is satis- 
fied, the complete trajectory is readily calculated. An 
important parameter of this motion is the distance 
parallel to the initial course from the point at which 
the rudder is thrown over to the center of the steady 



Figure 2. Diagram illustrating torpedo reach and 
method of calculation. 


turning circle. This ‘h'each” R of the torpedo is seen 
from the diagram in Figure 2 to be given approxi- 
mately by 


0 

R = x— p6 = s — — 


(13) 


Substitution of (11) into (13) gives 


R = 


(fll/pi + ^2/p2 ~ O'z) 


O, 


(14) 


which is independent of s, as, of course, it must be for 
the method of calculation to be valid. Equation (14) 
may be simplified with the help of (4) and (6) to 


Ck - Um2 - m) nCi -j- 
R = ^ r- • (Ih) 


Cm + XC; 


CiCk — rnCm 


Here X = CV Cx is the fraction of the torpedo length 
aft of the CG at which the effective rudder force is 
applied. It is interesting to note that R is independent 
of ^0 and hence of the turning circle radius p for a 
given torpedo. From (12) and (11), So is seen to be 


_ (0.45 - + 111 pi + ^2/p2) _R + 0.45 

So — : — . ftol 


9., 




Thus the accuracy criterion becomes 


— piSo = I pi| • (i? + 0.45p) ^ 2.3 ; (17) 


this does depend on p and hence on ^o. 

Substitution of the provisional torpedo parameters 
given in Section 9.1, into the above equations gives 
the following approximate numerical values for the 
Mark 13-2 torpedo with and without the shroud ring: 


Shroud Plain 

p (lengths) 88.5 17.7 

p(feet) 1,190 238 

(lengths) 1.61 4.22 

7?(feet) 21.6 56.7 

-piSo 23.2 2.80 


The accuracy criterion, which is barely satisfied by 
the plain tail torpedo, is very well met by the shroud 
tail torpedo. In general, a small value of R is asso- 
ciated with a large degree of dynamic stability; the 
more stable the torpedo, the more rapidly it ap- 
proaches a new state of steady motion. 


113 MANEUVERING: 

PULL-OUT FROM TURN 

As the second example, a solution of the equations 
of motion will be obtained for the case in which the 
rudder is suddenly set amidships while the torpedo is 
in a steady turn. The equations are now (1) with 
^0 = 0, from which it is seen that the solution is (2) 
with as = 9^ = 0. The coefficient ai, a 2 , 9i, and 92 
can be found as in Section 11.2 in terms of the initial 
conditions. It will be assumed that at s = 0, when 
the rudder is thrown amidships, the orientation of 
the torpedo is = 0 and the torpedo is circling to 
the left. Since in the steady turn the torpedo noses in 
toward the center of the turn, the initial value of a is 
negative and equal in magnitude to the as calculated 
from the first of equations (4). In similar fashion, the 
initial value of 9 is positive and equal to the 9s calcu- 
lated from the second of equations (4). The initial 
values of a' and 9' may be found from the equations 
of motion, as before. 

The solution for the course angle 6 is given by (10) 
with jSa = Oa = 0 so that asymptotically 6 approaches 
a constant value doo and the motion approaches a 


142 


STEERING CONTROL 


straight line. The trajectory may be found by inte- 
grating the approximate equations (9) : 

y{x) =-(^ + ^)x+ (gL + - 1) 

\Pl V2/ Pi 

-f (Q^2 + f^2./p2) — 1) 

P2 

where x has been substituted for s. The asymptotic 
form of this is the straight line: 

y(x) ►(9coX — 5. (19) 


y 



For the situation illustrated in the Figure 3, ao is 
negative and flo is positive. Since ao and Ho are pro- 
portional to the rudder throw ^0 in the initial turn 
(see equations 4), 5 and 6^ are also, and the quantity 
6/000 is independent of ^0 and hence of the initial 
turning radius. 

Numerical values may be obtained with the help of 
the approximate constants given in Section 9.1. 
Equations (20) then become 


000 — — 0.98q;o 0.4/3Ho , 

6 = — 3.79q:o + 0.505Ho (shroud) ; 

000 — — 5.45q!o “h 0.954Ho , 

8 = -28.9ao + 3.33Ho (plain) . (21) 


From the results of Section 11.2 

ao = —.0065 radian = —0.37°, 

Ho = .0113, (shroud) ; 
ao = — .0284 radian = — 1.63° , 

Ho = .0565 , (plain) . (22) 


Substitution of the torpedo parameters into the ex- 
pressions for ai, etc., gives 


— 1712.0 mao -f- wC;Ho 

CiCk - rnCm 


( 20 ) 


Substitution of (22) in (21) gives finally 



Shroud 

Plain 

000 

0.7° 

12.0° 

d 

0.030 

1.01 

d 00 

2.6 

4.8 


~ [Ck{OiCk — inCm) Cm(llCl + in2OK)]l7l2Oi0 
iCiCK - niCm)- 

[Ci{nCi-\-m2CK) — im2 — m){CiCK — mCm)]nilo 
{CiCk ~ mCmY 


8 and 8 da, are, of course, measured in lengths. It is 
apparent that the increase in stability that accom- 
panies the addition of the shroud ring greatly reduces 
the angle by which the torpedo overshoots the desired 
new heading. 


Chapter 12 

DEPTH CONTROL 


12.1 AVAILABLE PRINCIPLES AND 

METHODS 

I T IS NATURAL to tliiiik of the depth of a torpedo as 
being controlled simply by a h^Tlrostatic bellows 
or some similar pressure-indicating device. According 
to such a scheme, if the torpedo rises above set depth, 
the hydrostat calls for down elevator, and conversely. 
However, it is easwto see from the Xyquist criterion 
discussed in Section 9.1 that a proportional control 
based on this principle would be unstable. If h is the 
distance of the torpedo above set depth and ^ is the 



Figure 1. Xyquist diagram from curves in Chapter 9, 
Figure 4. 


elevator angle, the relation between the two quan- 
tities should be ^ = —ah, where cr is a positive propor- 
tionality constant. According to the curves of the 
phase of /?o/^o presented in Figure 4 of Chapter 9, this 
would give a Xyquist diagram of the form shown 
schematically in Figure 1. Since this encloses the 
unit point for any value of a, the system is unstable. 
In similar fashion, since the lag of h behind ^ is 
always greater than 180°, the discussion of Chapter 
10 shows that a two-position control based on hydro- 
stat indication alone will not give satisfactory opera- 
tion. 

From a physical viewpoint, it may be said that the 
hydrostat fails to give stable control because it does 
not anticipate incipient deviations from set depth 


before they develop to such a point that they cannot 
be handled. This lack of anticipation is equivalent 
mathematically to excessive phase lag (greater than 
180°). In order to get a stable control it is necessary 
to introduce an anticipatory device, which has the 
effect of decreasing the overall phase lag of the sys- 
tem. This can be done in a variety of ways, but in 
each case it is necessary to retain the hydrostat as 
part of the control mechanism in order that the tor- 
pedo may have an indication as to the depth at 
which it is supposed to run. Thus the problem of 
depth control during the steady run may be said to 
be the resolution of a conflict between too much 
anticipation and too little hydrostat on the one hand, 
which permits minor disturbances to give rise to 
excessive Avandering from set depth, and too little 
anticipation and too much hydrostat on the other 
hand, which leads to instability. This situation mani- 
fests itself both when proportional and two-position 
controls are used. 

The most widely used anticipatory device is the 
pendulum. The hydrostat is usually coupled to the 
pendulum through a linkage and the pendulum to the 
elevators through a depth engine. Thus the orienta- 
tion of the pendulum Avith respect to the torpedo, 
and hence the position of the elevators, depends both 
on the orientation angle (3 of the torpedo in space and 
on the force exerted on it by the hydrostat. Since the 
torpedo usually noses up before decreasing its depth, 
the pendulum, insofar as it indicates inclination of 
the torpedo, serves the desired purpose. The dia- 
grams in Figure 2 shoAv schematic Xyquist curves 
(derived from Figures 2 and 4 of Chapter 9) for a 
pendulum control alone, for A\hich ^ = —7/3, and for 
combined pendulum-hydrostat controls, for Avhich 
^ = — 7[ jS + ///(TF/F) 1 , AAdiere for the present pur- 
pose (3 is measured in degrees and h in feet. The IF/F 
ratio is the number of feet of depth change that is 
equivalent to 1° of change of inclination. It is clear 
from the diagrams that too small a IF I" leads to 
instability; as pointed out above, too large a IF/F 
suppresses the hydrostat effect so that the torpedo 
may be too insensitive to depth. TF/F values noAv in 
use range for the most part from 2 to 4. 

The foregoing discussion assumes that the system 



143 


144 


DEPTH CONTROL 



IDEAL PENDULUM ALONE 




Figure 2. Nyquist diagrams for various pendulum 
controls. 


is ideal in the sense that the hydrostat indicates 
depth, the pendulum indicates inclination, and the 
combined signal is transmitted without delay to the 
elevators. Actual depth controls fall short of this 
ideal in all three respects. The hydrostat indicates 
pressure rather than depth, and this varies from point 
to point over the torpedo and with its speed and 
pitch. Enough data are available on dynamic pres- 
sure distributions to take this into account in making 
the calculations. The effect is generally small and can 
be made negligible by suitably placing the hydrostat 
intake. 

The pendulum is acted upon both by gravity and 
accelerations and, moreover, possesses inertia which 
affects its displacement. The accelerations acting on 
the pendulum depend not only on the translational 
motion of the torpedo, but on its rotational motion as 
well since the pendulum is usually mounted some dis- 
tance from the CG of the torpedo. The effect of 
accelerations and inertia on the motion of a pendulum 
is best found by setting up the Lagrangian equation 
of motion in terms of the orientation angle x of the 
pendulum in space, when the torpedo range traveled 
r, height h, and orientation angle — /3 are specified 
functions of the time. Suppose that the pendulum 
pivot is mounted a distance xq aft of and yo above the 
torpedo CG, that L is the distance from its center of 
mass to the pivot, and that its mass and moment of 
inertia about the pivot are M and I. Then the equa- 
tion of motion is 

External torques MgL X sin % + 

(gravity) (inertia) 

-\- il/LfXcos X + il/L/?Xsin x + il/LX/3“ +il/LT/3, 

(longitudinal (vertical (centrifugal (angular 

acceleration) acceleration) acceleration) acceleration) 

( 1 ) 

where the origin of each term is indicated in paren- 
theses beneath it. Here = xq cos (x + iS) — yo 
sin (x + |S) is the distance from the torpedo CG to 
the pendulum along a line perpendicular to the pen- 
dulum, and Y = .To sin (x + i^) + 2/o cos (x + /3) is 
the distance from the pendulum pivot to the point of 
intersection of this line with the pendulum, measured 
positive when the pivot is above the intersection 
point. 

Equation (1) can be linearized by assuming that 
all angles are small. It is of interest to express the 
motion in terms of the angle = x ^ that the 
pendulum makes with the perpendicular to the tor- 
pedo axis since it is the orientation of the pendulum 


AVAILABLE PRINCIPLES AND METHODS 


145 


relative to the torpedo that provides a useable signal. 
The result is 

+ MgLrp = External torques — MLf — MLxof^ 

-[{I-MLY)i5 + MgL^] . (2) 

The first term on the left side is the inertia of the 
pendulum, and the second term is the gravitational 
restoring torque. Among the external torcpies may be 
a spring restoring torcpie proportional to which 
would modify the gravitational term and a viscous 
damping torque proportional to \p. The next two 


the ^ terms are of particular interest since the pendu- 
lum was originally introduced to provide a measure 
of It is evident that the effect produced by change 
of orientation depends in general on the rate or fre- 
quency of the angular motion of the torpedo. In par- 
ticular, there is a frequency of sinusoidal oscillation, 

g ~1 

C0.4 = , (4) 

U/{ML - }^)J ’ ^ ^ 

for which there is no pendulum motion contributed 
by the (3 terms. This is referred to as an antiresonance 


TORPEDO 



terms on the right side are the torcjues produced by 
longitudinal and vertical accelerations; the latter of 
these vanishes if the average orientation angle xo of 
the pendulum is zero (vertical in space). The last 
terms on the right side include the additional gravi- 
tational torcpie due to inclination of the torpedo, and 
the effect of angular acceleration of the torpedo. 

Suppose now that the hydrostat coupling (which 
appears as one of the external torques) is neglected 
for the moment, and a spring restoring torque Kxp 
and damping torcpie Rxp are introduced. Then ecpia- 
tion (2) becomes 

Ip l^P i^lgl^ “h IP)P 

= -^fLr-^^Lxoil-[(I-^ILY)i3-\-MgL(3] . (3) 

This may be thought of as a pendulum having an 
impedance characterized by the inertia, damping, 
and stiffness coefficients on the left side and driven 
by the forcing terms on the right side. Of these latter. 


frecpiency, in contrast with the resonance frequency 


to 72 


MgL + K 




/ 



which is the natural frecpiency of the pendulum and 
springs (provided the damping term R is not too 
large). The antiresonance frequency is equal in mag- 
nitude to the natural frequency of a simple pendulum 
the lengtli of which is etjual to the distance of the 
center of percussion of the pendulum below the point 
of intersection of the pendulum with a line through 
the torpedo CG perpendicular to the pendulum. If 
the center of percussion of the pendulum is above the 
intersection point, the antiresonance phenomenon 
does not occur. Considered from the point of view of 
the Nycpiist diagram, the resonance and antireso- 
nance can each introduce phase lags of the order of 
180°; since the former is associated with a large am- 
plitude and the latter with zero amplitude, the order 


146 


DEPTH CONTROL 


in which they occur as the frequency increases may 
be of importance in determining the stability of the 
control. This is discussed further in Section 12.2. 

It is clear, therefore, that a pendulum is far from 
an ideal device and can introduce additional phase 
lags and amplitude changes which may be (and 
generally are) undesirable. This has led to the inves- 
tigation of other methods for obtaining the antici- 
patory action necessary for satisfactory operation. 
The pendulum inertia and resonance may be elimi- 
nated by capturing the pendulum so that its motion 
is severely limited. A pneumatic proportional control 
has been developed along these lines by the Foxboro 
Corporation in which a feed-back link supplies a 
torque which is just sufficient to keep the pendulum 
from moving. The same torque operates the eleva- 
tors so that the system supplies a signal proportional 
to the right side of equation (3) and eliminates the 
impedance lag inherent in the left side. A very sim- 
ple captured pendulum has been used by the West- 
inghouse Electric and Manufacturing Company on 
some of their electric torpedoes. This is a two-posi- 
tion control in which the pendulum is permitted just 
enough motion to make or break an electric contact 
which supplies current through a relay to solenoids 
that move the elevators. While controls of this type 
remove the impedance lag, they cannot, of course, 
make the pendulum insensitive to accelerations. 

For some applications it is important to have a 
depth control which is unresponsive to accelerations. 
Controls of this type have been constructed by the 
United Shoe Machinery Corporation, using a gyro- 
scope as a standard of orientation in the vertical 
plane. An uncorrected gyro is unsatisfactory for this 
purpose, even though it can be used for steering, 
since the limitations on depth are far more severe 
than those on deflection. The USMC controls have 
therefore evolved along two lines. First, the gyro is 
processed by friction so that it tends to be oriented 
perpendicular to the torpedo axis. In this arrange- 
ment the gyro corrects for small deviations from 
running orientation angle, and the hydrostat adjusts 
the average depth. Second, the primary anticipation 
is provided by a rate of change of depth indicator, 
and additional stability is obtained from a spring- 
captured gyro which indicates rate of change of 
orientation angle. Although this latter system has 
not as yet been tried, analysis indicates that it should 
be successful with reasonable values of the param- 
eters. The hydrostat-rate of depth control without 
the gyro had been tried earlier by Foxboro and pre- 


liminary results were unsatisfactory. The computed 
margin of stability turns out to be very small with 
such a control, particularly with the plain tail Mark 
13-2, which is somewhat less stable than the present 
shroud tail torpedo. Addition of the captured gyro 
should make this a practical device. 

Thus far, very little has been said about the trans- 
mission of the depth-control signal to the elevators. 
This is the third point at which the overall system can 
deviate from the ideal. All American torpedoes with 
the exception of some of those under development by 
Westing-house use proportional depth controls and 
therefore require a depth engine which will transmit 
the desired indication without delay or change in 
amplitude and with sufficient force output to operate 
the elevators. Such proportional engines may be 
grouped into two classes: force-proportional engines 
and positioning- engines. Force-proportional steering 
controls were discussed in Section 11.1. A pneumatic 
device of the type described there has also been de- 
veloped by Columbia University and the American 
Can Company for the depth control of the Mark 25 
torpedo. A force-proportional electric control has 
been used experimentally by Westinghouse. This 
control has a multi-contact unit (Silverstat) which 
changes the current through the elevator solenoids in 
discrete steps as the pendulum orientation changes; 
as with the pneumatic control, the elevators are 
loaded with springs to insure proportional action. 

The standard depth engine used until recently on 
all American torpedoes is of the positioning or dis- 
placement-proportional type. In this engine any 
difference in displacement of the piston and the valve 
stem, which is attached to the pendulum, produces 
an unbalanced air force on the piston which tends to 
bring it into line with the valve. It suffers from the 
drawback that, due to close machining tolerances, it 
is difficult to manufacture and adjust and, unless 
maintained in optimum condition, will not respond to 
rapid motions of the pendulum. An improvement on 
this has recently been developed in conjunction with 
the blade type force-proportional control for possible 
use on the Mark 25 torpedo. In this device the pendu- 
lum carries a jet which feeds high pressure air into 
one or the other of two receptive orifices, causing the 
pendulum support to move until whatever force is 
being exerted on the pendulum is balanced by a 
spring- and the pendulum remains in its neutral 
orientation. The pendulum support is rigidly con- 
nected to the elevators and hence displaces them in 
proportion to the pendulum torque. 


EXAMPLE: PENDE LUM-HVDROSTAT PROPORTIONAL CONTROL 


147 


All of these enjiines suffer to a j’reater or lesser 
extent from phase la^s, and these impair the opera- 
tion of both proportional and two-position systems. 
Such laji'S tend to make the former unstable (see 
Section 9.2) and increase the period of the latter until 
the amplitude of the depth wave is excessive (see 
Chapter 10). The best way in which to reduce such 
lags is by careful design, construction, and main- 
tenance. In pneumatic controls, connecting air lines 
should l)e as short as possible, high pressure air 
should be used, and provision should be made for 
exhausting as well as filling engine cylinders. In 
electric controls, relay operation should be made as 
rapid as possible and time delays due to current 
build-up in solenoids should be reduced by proper 
design. In two-position controls in particular, phase 
advance schemes have been employed with consider- 
able success to partially compensate for time delays 
and other phase lags that cannot be eliminated. These 
may involve spring-dashpot systems mounted on the 
pendulum which add derivatives of the pendulum 
displacement to the signal supplied to the depth 
engine. Perhaps the most successful attempt along 
these lines is that employed by Westinghouse in one 
of their two-position electric controls. The pendulum 
is allowed to swing nearly freely and a contact-relay 
system is set up so that the elevators are thrown up 
when the pendulum leaves one of two contacts spaced 
a finite angle apart, and thrown down when it leaves 
the other. Although the relay system is more compli- 
cated here than in the single-contact captured pen- 
dulum, the gain from the advanced phase at which 
elevator action takes place more than compensates 
for the slight increase in relay time delay. 

Two effects which have not as yet been given the 
attention which they deserve from an analytical 
point of view are Coulomb or static friction and free 
play. Static friction can affect the motion of the 
pendulum, or the engine motion directly in a force- 
proportional system. Free play can occur in the pen- 
dulum-hydrostat linkage or in the engine output. 
Both of these are non-linear effects and hence diffi- 
cult to take into account without great theoretical 
complications. It seems, however, that both of these 
effects will introduce additional phase lags into any 
control, and this is their principal effect in two-posi- 
tion controls. In proportional controls they may also 
produce oscillations, not necessarily periodic in char- 
acter, of sufficient amplitude to take up the frictional 
force or the displacement free play. In well-designed 
systems these oscillations are probably too small to 


observe in the overall motion of the torpedo, but 
might be detected with suital)le instrumentation. 

12.2 EXAMPLE: PENDULUM-HYDROSTAT 
PROPORTIONAL CONTROL 

As an illustrative example of a depth control, a 
simple pendulum-hydrostat proportional system will 
be considered in this section with the help of the Xy- 
quist criterion. It will be assumed that the pendulum 
is mounted vertically in space when the torpedo is at 
its running orientation angle so that xo = 0. Since 
the drag and propeller thrust depend only very 
slightly on attack angle, equation (4) of Chapter 7 
indicates that the longitudinal acceleration is zero. 
Then the pendulum equation (3) becomes 

-h /fiA + + K)^p 

= _[(/_ MLY)ii + Mym + , (6) 

where a hydrostat torque has been added correspond- 
ing to a particular IT T" ratio; angles are measured in 
degrees and depth in feet. It will be assumed that the 
depth engine is an ideal one of the positioning type 
so that 

^ = a\J/ . ( 7 ) 

In using the Nyquist method, it is necessary to 
find the steady response of the pendulum to oscillat- 
ing j8 and h of angular frecpiency co (/3 = etc.). 
In the notation of Section 9.2, equation (6) gives 

_ + lMgL-{I-MLy)oi^]^o+[MgL/(]V/V)]h 
{MgL + K) - /a;2 -h iRo, 

(8) 

With the substitutions 

_ a MgL 

“ M(jL + K ’ 

2x7? 

^ “ MgL + K ’ 

(9) 

co.-l 


OiR 


148 


DEPTH CONTROL 


where coa and cor are defined by (4) and (5), equation 
(7) becomes 

. * ^ _ y[-m-v^/vA^) + ho/(W/V)] _ 

[ 1 — v~/vr~ + ^ pr ] 

This may be expressed in terms of the torpedo- 
response functions plotted in Figures 1, 2, 3, and 4 of 
Chapter 9 to give an overall response function for the 
torpedo plus control : 

^ 0 * _ 7[ (/3o (1 - vrh/-) + (/?o/^o)/(TF.aO ] 

y 1 ■’ / I * \ / 

lo 1 — y-/ Vr- I pV 

Current practice with the Mark 13-2 torpedo 
favors values close to 7 = 3, W /V = 3; vr is gener- 
ally about 1 c, and va may vary over a wide range. It 
is clear then from a comparison of Figures 1 and 3 of 
Chapter 9 that for a considerable frequency range 
around 1 c, which corresponds to = 1.3 at 40 knots, 
the magnitude of (3o/^o is much greater than the mag- 
nitude of (ho '^o) '{W/V). Thus the /3 term is the 
dominant one, except near va, where it vanishes be- 
cause of the antiresonance. For moderate values of 
the damping coefficient p, the resonance will cause 
the overall response to become large near pr. Now 
both the resonance and the antiresonance have phase 
lag increases of the order of 180° associated with 
them. Thus the structure of the Nyquist curve is 
altered in a fundamental way as va is changed from 
a value less than to a value greater than vr. This 
is illustrated schematically in Figure 4. In each case, 
the curve passes close to the origin near the anti- 
resonance frequency and describes a large semicircle 
near the resonance frequency. It follows that the sys- 
tem is stable when va<vr and unstable when va>vr. 

The standard IMark 13-2 depth control appears to 
fall in the latter class, with va slightly greater than 
Vr. However, the pendulum damping is large, and 
this has the effect of reducing the magnitude of the 
resonance peak. A careful computation, which is 
based on the best available parameters for both tor- 
pedo and control and which considers other effects 
omitted in the discussion of this section (dependence 
of hydrostat pressure on attack angle and the finite- 
ness of xo), shows that the plain tail torpedo is just 
on the edge of instability at 40 knots. Reduction of 




Figure 4. Variation of Xyquist curves with the 
change in the va (pendulum antiresonant frequency), 

VR (pendulum resonant frequency) relationship. 

the speed to 333^ knots, or addition of the shroud 
ring, or both, serve to put the torpedo well into the 
region of stable operation. Reduction of speed re- 
duces the phase lag and increases the gain, but the 
former effect is more important than the latter; addi- 
tion of the shroud ring decreases both phase lag and 
gain. Thus both of these changes improve the sta- 
bility. 



Chapter 13 

POWER PLANT 


W HEX THIS REPORT was first projected, it was 
planned to include a discussion of torpedo 
power plants. However, while this report was being 
prepared the United States Navy Coordinator of 
Research and Development requested the National 
Defense Research Committee to undertake a tor- 
pedo surve}" under Project X-121. A panel of experts 
was assembled, and among the subjects upon which 
a report was presented was torpedo power plants. 

As this report is presumabh^ available to interested 
parties through the Office of Naval Research and is 
quite comprehensive, it is not thought necessary to 
present here in a degree the same material. 

For additional detailed information on torpedo 
power plants the reader is referred to the Bibliog- 
raphy. 




149 


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BIBLIOGRAPHY 


Xumbers such as Div. 6-SOO-Ml indicate that the document listed has been microfilmed and that its title appears in 
the microfilm index printed in a separate volume. For access to the index volume and to the microfilm, consult the 
Army or Xhavy agency listed on the reverse of the half-title page. 


GENERAL 

1. Selected Index of British Reports on Torpedoes, Richard H. 

Bolt, CUDWR,® March 1944. Div. 6-800-Ml 

2. Program Analysis of the Torpedo Division, OEMsr-287, 

HUSL,b May 30, 1944. Div. 6-800-M2 

3. Conference on Program Analysis of the Torpedo Division 

of Harvard Underwater Sound Laboratory Held in the New 
York Division Six Office, June 5, 1944, Richard H. Bolt, 
June 1944. Div. 6-800-M3 

4. Report on Studies Made in Connection with Projects AC-70 
and NO-177, William V. Houston, OSRD 53, NDRC 
6.1-srll31-1182, CUDWR, Oct. 25, 1944. Div. 6-800-M5 

5. A Theoretical Study of the Effectiveness of a 20-Knot 
Acoustic Torpedo and of Possible Modifications Having 
Lower Speeds With or Without an Automatic Speed-Chang- 
ing Mechanism, Conyers Herring and E. Ward Emery, 
NDRC 6. 1-srl 131-1882, CUDWR, Nov. 22, 1944. 

Div. 6-912.4-Ml 

6. Summary Report of the CIT Morris Dam Group of Division 
3 NDRC, L. B. Slichter and others, CIT,® 1945. 

HYDRODYNAMICS AND AERODYNAMICS 

7. Hydrodynamics, H. Lamb, 4th Edition, 1916, Chap. VI, 
pp. 124, 127. 

8. Report of Investigation of Torpedoes with Screws Running, 
Conducted in the Aerodynamics Department, National 
Physical Laboratory, British, January 1928. 

9. Analysis of Aircraft Torpedo Drops, The Effect of Entry 

Conditions and U nderwater Roll on the Initial Dive, L. W. 
Parkin and K. D. Tocher, OSRD WA-1241-4, Report 
TDU/18/1943, Torpedo Development Unit, RAF Sta- 
tion, Gosport, August 1943. Div. 6-810.21-M3 

10. The Underwater Behavior of the 11.75-in. Aircraft Rocket, 

1. S. Bowen, OEMsr-418, Division 3 Report IPC 65, 
CIT, Oct. 25, 1944. AMP-406-M4 

11. Tests of Darts at the New London Escape Tower, L. J. 
Hooper, Report Dl/1318, X"LL,^ Sept. 26, 1941. 

Div. 6-810.22-Ml 

12. Free Fall of Streamlined Bodies in Water, R. G. Folsom 
and Morrough P. O’Brien, UCDWR,® Oct. 22, 1941. 

Div. 6-810.2-Ml 


^ Columbia University Division of War Research. 

^ Harvard Underwater Sound Laboratory. 

® California Institute of Technology. 

New London Laboratory. 

® University of California Division of War Research. 


13. A Preliminary Study of the Ballistics of a Submerged Pro- 
jectile, Report 603, CIT, Oct. 23, 1941. 

Div. 6-810.2-M2 

14. Free Fall Tests of 1" x 6" Brass Darts at the Alden Hydrau- 

lic Laboratory, Worcester Polytechnic Institute, Worcester, 
Massachusetts, L. J. Hooper, Laboratory Report D1 /1983, 
NLL, Feb. 10, 1942. Div. 6-810.22-M2 

15. High Speed Photography of 1" x 6'' Darts at the Alden 
Hydraulic Laboratory, Worcester Polytechnic Institute, 
Worcester, Massachusetts, L. J. Hooper, Laboratory Re- 
port Dl/1985, NLL, Feb. 16, 1942. Div. 6-810.22-M3 

16. Summary, Coefficients of Drag and Terminal Velocities, 

Models of Underwater Projectiles, Report P16/2206, NLL, 
Mar. 10, 1942. Div. 6-810.2-M3 

17. The Trajectory of Stable Underwater Projectiles for the Case 
in which the Trajectory is Nearly Vertical and the Projectile 
is Traveling at Terminal Velocity, G. A. Gongwer, Labora- 
tory Report DlO/2258, NLL, May 1, 1942. 

Div. 6-810.21-Ml 

18. Investigation of the Control in Air of a Torpedo with Fixed 
Tailplane and Gyro-Controlled Ailerons, L. W. Parkin, 
OSRD WA-1 378-26, Torpedo Development Unit, July 

1942. Div. 6-810. 1-Ml 

19. British Studies of Hydrodynamics of Projectiles Discussion 
with Dr. V. 0. Knudsen, D. P. Fullerton, Report G2/3873, 
NLL, Sept. 1, 1942. 

20. Memorandum on Water Tunnel Tests of 2" Diameter Pro- 
jectiles with Hemispherical Noses and Square Ends, Robert 
T. Knapp, HML Report ND-10, CIT, Nov. 10, 1942. 

Div. 6-722.7-Ml 

21. Initial Underwater Torpedo Trajectories After Dropping, 
C. H. Tindal, NDRC 6.1-sr287-778, HUSL, May 25, 

1943. Div. 6-810.21-Ml 

22. Aircraft Torpedo Trajectory, U. S. Navy Torpedo Sta- 
tion, Newport, Rhode Island, October 1943. 

23. Selected Index of British Reports on Water Entry and 

Water Travel, William V. Houston, Project No. 176, 
CUDWR, November 1943. Div. 6-810.2-M4 

24. Selected Index of California Institute of Technology and 
Miscellaneous Reports on Water Entry and Water Travel, 
William V. Houston, CUDWR, November 1943. 

Div. 6-810.2-M5 

25. Relationship between Cavitation Bubble and Entrance Air 
Bubble of Torpedoes and Other Projectiles (Memorandum), 
Robert T. Knapp, OEMsr-207, CIT, Nov. 6, 1943. 

Div. 6-810.23-Ml 


152 


BIBLIOGRAPHY 


26. Observations on the Water Entry of a Torpedo (Local Inter- 

mediate Report), R. W. Ager, Report IOC-14, CIT, 
Xov. 23, 1943. Div. 6-810.21-M4 

27. Wind T unnel T ests Prior to Dropping Full Sized T orpedoes, 
Harvey A. Brooks and Carl i\I. Herget, HUSL, Dec. 21, 

1943. Div. 6-810.1-M2 

28. Preliminary Remarks on the Cavity JMade by a Projectile 
Entering Water, B. D. Blackwell, OSRD WA-1927-3d, 
Report SRE/UT/4, Department of Scientific Research 
and Experiment, Great Britain, February 1944. 

A:\IP-401.2-M4 

29. Ricochet off Water, Garrett Birkhoff, George D. Birkhoff, 

and others, OEMsr-1007, AMP^ Memorandum 42. 4M, 
A:\IG-C 157, May 1944. AMP-401. l-:\15 

30. The 24-ft- Wind Tunnel Tests on Four Torpedo Air Tails, 

P. J. Pearsall and T. B. Owen, OSRD \YA-2764-3, Tech- 
nical Note Aero. 1442, Royal Aircraft Establishment, 
Farnborough, May 1944. 6-810. 1-M3 

31. A Method of Prediction of the Upturning Undenvater Tra- 

jectories of Rockets in Two Dimensions, B. D. Blackwell, 
Report SRE/UT/8, LTBRC-31, Department of Scientific 
Research and Experiment, Admiralty, Great Britain, 
June 1944. AMP-406-M7 

32. hnpact Forces, Garrett Birkoff, OEYIsr-1384, AYIP 
Memorandum 42. 6M, AMG-Harvard, September 1944. 

A:\IP-404-M1 

33. Prediction of U nderwater Behavior from Wind Tunnel 
Measurements, R. A. Shaw, OSRD \VA-3024-14C, H/ 
Arm/ Res. 23, Ylarine Aircraft Experiment Establish- 
ment, Helensburgh, Great Britain, Sept. 26, 1944. 

AMP-401-M6 

34. Visit to California Institute of Technology, August 27- 

September 3, 1944, G. F. Wislicenus, Project NO-176, 
CUDWR, Sept. 9, 1944. Div. 6-800-M4 

35. Theoretical Depth Trajectories 149-B (YIemorandum), 

Harvey A. Brooks and Nelson M. Blachman, HUSL, 
Oct. 31, 1944. Div. 6-810.21-YI5 

36. Drag Coefficients of Spherical and Coniccd Motion Under- 
water, R. \V. Duncan and F. E. Bradley, OSRD WA- 
3519-2, Technical Note Arm. 301, Royal Aircraft Estab- 
lishment, Great Britain, November 1944. 

AMP-401.3-M4 

37. Information on Mk 13 Dynamics, Harvey A. Brooks, 

HUSL, Nov. 13, 1944. Div. 6-820. 1-M2 

38. The Effect of Trajectory Angle and Pitch Angle on the 
Initial Underwater Trajectory of the Torpedo, Ylarvin 
Gimprich, NDRC 6. 1-srl 131-1881, CUDWR, Nov. 18, 

1944. Div. 6-810.21-M6 

39. Tunnel Characteristics and Test Procedure for Bodies of 
Revolution. Cavitation Tests in the Iowa Variable- Pressure 
Water Tunnel, Hunter Rouse, John S. McNown, and 


En-Yun-Hsu, OEMsr-1353, Informal Report to Section 
12.1, State University of Iowa, Nov. 30, 1944. 

Div. 6-810.23-M2 

40. Moment of Inertia Characteristics of Mark-13 Type Tor- 
pedoes, Memorandum to Chief, Bureau of Aeronautics 
Section, Naval Torpedo Station, Newport, R. I., Dec. 1, 
1944. 

41. Cylindrical Body with Hemispherical Head, Cavitation 

Tests in the Iowa Variable-Pressure Water Tunnel, Hunter 
Rouse, John S. YIcNown, and En-Yun-Hsu, OEMsr- 
1353, Informal Report to Section 12.1, State Lhiiversity 
of Iowa, Dec. 4, 1944. Div. 6-810.23-YI3 

42. Cylindrical Body with Blunt Head, Cavitation Tests in the 

Iowa Variable-Pressure Water Tunnel, Hunter Rouse, 
John S. McNown, and En-Yun-Hsu, OEMsr-1353, In- 
formal Report to Section 12.1, State University of Iowa, 
Dec. 8, 1944. Div. 6-810.23-M4 

43. Head Studies, J. H. Wayland, Report TLP 28, TL-D 351, 
CIT, January 1945. 

44. Initial Dive and Recovery of Aircraft Torpedoes, Leonard 
1. Schiff, Project NO-176, CUDWR, Feb. 14, 1945. 

Div. 6-810.21-YI7 

45. U nderwater Performance of One Inch Diameter Models of a 

Family of Cone Head Rockets, R. A. Shaw and P. E. 
Naylor, OSRD WA-3996-9, Report H /Arm Res. 25, 
Marine Aircraft Experimental Establishment, Great Brit- 
ain, Feb. 7, 1945. AMP-406-M6 

46. Heat Treatment of Standard Mk 13 Torpedo Propellers to 
Withstand Entry Impact Forces, W. Harry Johns, Jr., 
NDRC 6. 1-srl 131-1886, CUDWR, Feb. 8, 1945. 

Div. 6-810.2-M7 

47. Cavitation Tests on a Systematic Series of Torpedo Heads. 
Water Tunnel Characteristics and Test Procedure, Hunter 
Rouse, John S. McNown, and En-Yun-Hsu, NDRC 
6.1-srl353-2190, State University of Iowa, Institute of 
Hydraulic Research, Feb. 22, 1945. Div. 6-810. 23-M5 

48. The Mechanism of Pitch Sensitivity of Aircraft Torpedoes, 
Harold Wayland, Report NOC-47.1, CIT, Feb. 27, 1945. 

Div. 6-810.1-M4 

49. Cavitation Tests on a Systematic Series of Torpedo Heads, 

Hemispherical Head, Hunter Rouse, John S. YIcNown, 
and En-Yun-Hsu, OSRD 5059, NDRC 6.1-srl353-2191, 
State University of Iowa, Institute of Hydraulic Research, 
Feb. 28, 1945. Div. 6-71 2-M5 

50. Cavitation Tests on a Systematic Series of Torpedo Heads, 

Blunt Head, Hunter Rouse, John S. McNown, and En- 
Yun-Hsu, OSRD 5056, NDRC 6.1-srl353-2192, State 
University of Iowa, Institute of Hydraulic Research, 
Mar. 5, 1945. Div. 6-712-YI6 

51. Cavitation Tests on a Systematic Series of Torpedo Heads, 

1 14 Caliber Rounded Head, Hunter Rouse, John S. Mc- 
Nown, and En-Yun-Hsu, NDRC 6.1-srl353-2193, State 
University of Iowa, Institute of Hydraulic Research, 
Mar. 10, 1945. Div. 6-810.23-M6 


^ Applied Mathematics Panel. 




BIBLIOGRAPHY 


153 


oJ. 


53. 


54. 


55. 


56. 


57. 


58. 


59. 


60. 


61. 


62. 

63. 


64. 


g 


Cavitation Tests on a Systetnatic Series of Torpedo Heads, 
1/S Caliber Rounded Head, Hunter Rouse, John S. AIc- 
Xown, and En-Yun-Hsu, XDRC 6.1-srl353-2194, State 
University of Iowa, Institute of Hydraulic Research, 
Mar. 15, 1945. Div. 6-810.23-M7 

Cavitation Tests on a Systematic Series of Torpedo Heads, 

1- Caliber Ogival Head, Hunter Rouse, John S. McXown, 

and En-Yun-Hsu, OSRD 5055, X"DRC 6.1-srl353-2195, 
State University of Iowa, Institute of Hydraulic Re- 
search, Mar. 20, 1945. Div. 6-712-I\I8 

Cavitation Tests on a Systematic Series of Torpedo Heads, 

2- Caliber Ogival Head, Hunter Rouse, John S. McX^^own, 

and En-Yun-Hsu, OSRD 5054, X^DRC 6.1-srl353-2196, 
State L’niversity of Iowa, Institute of Hydraulic Research, 
Mar. 26, 1945.^ Div. 6-712-M9 

Analysis of Aircraft Drops of Mark 13-2A and Mark 13-3 
Torpedoes fitted with Eight Degree Cone Angle Shroud Ring 
Tail, K. H. Keller, Dec. 14, 1944. In: Analysis of Aircraft 
Launchings of Torpedoes Equipped with Shroud Rings, 
Karl H. Keller, Marvin Gimprich and lY. H. Wilson, 
XDRC 6. 1-srl 131-1887, Project XO-176, CUDWR, 
Mar. 26, 1945. Div. 6-810. 1-:H5 

Summary of Cavitation Tests on a Systematic Series of Tor- 
pedo Heads (Final Report), Hunter Rouse, John S. 
McX^own, and En-AYn-Hsu, XDRC 6. 1-sr 1353-2330, 
State Lhiiversity of Iowa, Institute of Hydraulic Re- 
search, May 31, 1945. Div. 6-810.23-M8 

Water Entry Bibliography, OEMsr-1384, AMP Alemo- 
randum 42. 8M, AMG-H 11, June 1945. AMP-401-M10 

Values of Lift and Moment Coefficient Derivatives for the 
American Mark 13 Aircraft Torpedo, B. G. X^eal, OSRD 
WA-4557-3, Report SRD/UT/9, Department of Scien- 
tific Research and E.xperiment, Great Britain, June 1945. 

Div. 6-810. 1-M6 

A Study of the Depth of Initial Dive of a Mark-13 Torpedo 
with Shroud Ring, Alarvin Gimprich, OSRD 5244, XDRC 

6. 1- srl 131-1889, June 9, 1945. AMP-405. 3-M8 

Hydrodynamic Forces Resulting from Cavitation on Under- 
Water Bodies, James W. Daily, OSRD 5756, X^DRC 

6.1- sr207-2242, Division 6 Laboratory Report XD-31.2, 

MIT,g July 21, 1945. AMP-401. 5-M4 

Attitude of Torpedoes Released from Airplanes (Interim 
Report), Marvin Gimprich, X"DRC 6. 1-srl 131-1893, 
Dec. 21, 1945. Div. 6-810.1-M7 


COXTROL OF UNDERWATER RUN 


Bell System Technical Journal, Harry X'yquist, Vol. II, 
1932, p. 126. 

Differentialgleichungen der Fhysik, Frank and von Mieses, 
Yol. I, p. 163. 

Determination of Running Depth of Test Torpedoes by a 
Sonic. Method, Donald A. Proudfoot, Memorandum for 
hie G/10/R165, XLL, Fah. 18, 1943. Div. 6-820. 22-Ml 

Mas.sachusetts Institute of Technology. 




. 4- 




65. Analysis of Search and Pursuit Patterns Employing Me- 

chanical Depth Control, H. Poritsky and L. J. Savage, 
XDRC 6.1-srl 131-1 155, Project XO-181, CUDWR, Dec. 
22, 1943. Div. 6-820.22-M2 

66. Memorandum on Torpedo Steering. William V. Houston 

[CUDWR], Feb. 14, 1944. Div. 6-820.21-Ml 

67. Steering Control of the Mark 13-2 Torpedo, Leonard 1. 
Schiff, Project XO-176, CUDWR, May 29, 1944. 

Div. 6-820.2 1-M2 

68. Proportioned and On-Off Control Systems, J. C. Lozier, 

Report 44-35 10-JCL-GH, Bell Telephone Laboratories, 
Aug. 15, 1944. Div. 6-820.1-Ml 

69. Depth Turning Radius of Mark 13 with Shroud Rings in 

Forward Position, Gilford G. Quarles, HUSL, Oct. 30, 
1944. Div. 6-820.22-AI3 

70. Supplement to “Depth-Turning Radius of Mark 13 with 

Shroud Ring in Forward Position” (Memorandum), Gil- 
ford G. Quarles, HUSL, Oct. 31, 1944. Div. 6-820. 22-M4 

71. Hydrodynamic Properties of the JMark 13-2 and 13-2A 
Torpedoes with Plain and Shroud Ring Tails, Leonard I. 
Schiff, Project X'^0-176, CUDWR, N^ov. 7, 1944. 

Div. 6-810.2-M6 

72. Development of Depth and Steering Controls for the Mark 25 
Torpedo, William V. Houston, N^DRC 6. 1-srl 131-2348, 
Project XO-176, CUDWR, July 12, 1945. 

Div. 6-820- All 

POWER PLANT 

73. Preliminary Experiments with Mixtures of Tetranitro- 

methane and Iso-Octane, OEAIsr-124, Report 607, CIT, 
Xov. 20, 1941. Div. 6-830. 2- All 

74. Studies of the Restricted Burning of Certain Heterogeneous 
and Colloidal Propellants, OEAIsr-124, Report 609, De- 
partment of Chemical Engineering, CIT, Apr. 15, 1942. 

75. Study of Utilization of Tetranitromethane and Gasoline as 

Fuel in Jet Propulsive Equipment, OEAIsr-124, Report 
610, Department of Chemical Engineering, CIT, Apr. 
22, 1942. Div. 6-830.2-AI3 

76. Dynamometer Test of Aircraft Torpedoes, George Farnell 
and Ascher H. Shapiro, Project XO-176, Research Pro- 
ject DIC-6228, Report D-1, AIIT, Alay 23, 1944. 

Div. 6-830-AIl 

77. Dynamometer Tests of Aircraft Torpedoes, George Farnell 
and Ascher H. Shapiro, Project X" 0-176, Research Pro- 
ject DIC-6228, Report D-2, AIIT, June 30, 1944. 

Div. 6-830-AIl 

78. Torpedo Igniter Type B Developed by Remington Arms 

Company, Inc., A. E. Buchanan, Jr., X'^DRC 6. 1-srl 131- 
1851, prepared by Remington Arms Co., Inc., for Special 
Studies Group, Sept. 5, 1944. Div. 6-830.1-AIl 

79. Thermodynamic Analysis of the Combustion of Ethyl Alco- 

hol, John A. Goff, Project X^O-176, XDRC 6.1-srl 131- 
1847, CUDWR, Sept. 18, 1944. Div. 6-830.2-AI4 






154 


BIBLIOGRAPHY 


80. Gas Generating Systems for Torpedoes. An Experimental 
Study of Systems Using Ethyl Alcohol and Air for Combus- 
tion and Either Water or Ethyl Alcohol as a Coolant, George 
Farnell, Chas. S. Hofmann, Don G. Jordan, William A. 
Reed, Dumont Rush, and Ascher H. Shapiro, NDRC 

6.1- srll98-2111, Project NO-176, Research Project DIC- 
6228, Report R-1, MIT, Nov. 4, 1944. Div. 6-830.2-M5 

81. Gas Generating Systems for Torpedoes. An Experimental 

Study of the Mark EX-25-0 Combustion Pot Using Ethyl 
Alcohol and Air for Combustion and Ethyl Alcohol as a 
Coolant, George Farnell, Chas. S. Hofmann, Don G. 
Jordan, William A. Reed, Dumont Rush, and Ascher H. 
Shapiro, NDRC 6.1-srll98-2112, Project NO-176, Re- 
search Project DIC-6228, Report R-2, MIT, Jan. 11, 
1945. Div. 6-830.2-M6 

82. Progress Report on Decomposition Studies (Memorandum 
to: Glenn C. Williams, Ernest P. Neumann, and Howard 
S. Gardner), Charles N. Satterfield and Wilburn H. Hoff- 
man, OEMsr-1289, Project NO-236, MIT, Jan. 30, 1945. 

Div. 6-830.21-Ml 

83. An Explorative Study of the Combustion of Nitropropane 
with Air and of Nitroethane with Air, William A. Reed and 
Ascher H. Shapiro, Project NO-176, Research Project 
DIC-6228, Report C-1, MIT, Feb. 13, 1945. 

Div. 6-830.2-M7 

84. Therjnodynamic Analysis of the Energy Producing Capa- 
bilities of Hydrogen Peroxide, Harold S. Mickley, NDRC 

6. 1- sr 1289-2 118, Project NO-236, Research Project DIC- 
6249, Report T-1, MIT, Mar. 28, 1945. Div. 6-830. 2-M8 

85. Torpedo Fuels, G. C. Williams and E. P. Neumann, 
NDRC 6.1-srl829-2119, Project NO-236, Research Pro- 
ject DIC-6249, Progress Report A-1, MIT, Apr. 11, 1945. 

Div. 6-830.2-M9 


86. Torpedo Fuels, G. C. Williams and E. P. Neumann, 
NDRC 6.1-srl289-2119, Project NO-236, Re.search Pro- 
ject DIC-6249, Progress Report A-2, MIT, May 14, 1945. 

Div. 6-830.2-M9 

87. Thermodynamic Analysis of the Energy Producing Capa- 
bilities of Hydrogen Peroxide, NDRC 6. 1-sr 1289-2 118, 
Project NO-236 and Research Project DIC-6249, Pro- 
gress Report T-2, MIT, June 14, 1945. Div. 6-830.2-M8 

88. Decomposition of 50 Weight Per Cent Hydrogen Peroxide 
Solution Using Calcium and Sodium Permanganate, 
Charles N. Satterfield and Wilburn H. Hoffman, NDRC 

6.1- srl289-2336, Project NO-236, Research Project DIC- 
6249, Progress Report C-1, MIT, June 28, 1945. 

Div. 6-830.21-M2 

89. Torpedo Igniter Design and Pyrotechnics Investigation, J. 

P. Catlin, W. L. Finlay and T. B. Johnson, NDRC 6.1- 
srl 131-2333, Aug. 31, 1945. Div. 6-830.1-M2 

90. An Experimental Investigation of Torpedo Power Plants 
(Final Report), C. Richard Soderberg and Ascher H. 
Shapiro, OSRD 6348, NDRC 6. 1-sr 1198-2385, Research 
Project DIC-6228, MIT, Aug. 31, 1945. 

Div. 6-830-M2 

91 . Investigation of T orpedo Fuels — An Experimental Study of 
the Peroxide-Ethanol Cycle (Final Report), NDRC 6.1- 
srl289-2391, MIT, Nov. 14, 1945. Div. 6-830.2-M10 

92. Design of the Mark 25 Torpedo (Section 2 of the Final 
Technical Report), OEMsr-1131, OSRD 6673, NDRC 

6. 1- srl 131-2393, Project NO-176, Dec. 31, 1945. 

Div. 6-800-M6 

93. Sea-Water Batteries (Final Report), OSRD 6420, NDRC 

6.1- srl069-2128, Bell Telephone Laboratories, Nov. 30, 

1945. Div. 6-647-Ml 


CONTRACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACTS 


Contract 

Xurnber 

Xanie and Address 
of Contractor 

Subject 

OEMsr-20 

The Trustees of Columbia University 
in the City of New York 

New York, New York 

Studies and e.xperimental investigations in connection with 
and for the development of equipment and methods per- 
taining to submarine warfare. 

OEMsr-1131 

The Trustees of Columbia University 
in the City of New AMrk 

New York, New AMrk 

Conduct studies and investigations in connection with the 
evaluation of the applicability of data, methods, devices, 
and systems pertaining to submarine and subsurface 
warfare. 

OEMsr-207 

California Institute of Technology 
Pasadena, California 

Construction and operation of a high-speed water tunnel, 
and use of such water tunnel in research and experimental 
investigations involving underwater projectiles and de- 
tection equipment. 

OEMsr-1105 

American Can Company 

New York, New York 

Conduct studies and experimental investigations in connec- 
tion with (i) the modification and improvement of torpedo 
design, with the general purpose of (a) enabling torpedoes 
to be dropped from aircraft without damage at higher 
speeds than is now possible and (b) improving the oper- 
ating characteristics of torpedoes designed for high under- 
water speed; and (ii) the construction of experimental 
models of torpedoes or parts thereof for test purposes. 

OEMsr-1198 

Massachusetts Institute of Technology 
Cambridge, Mass. 

Conduct studies and experimental investigations in connec- 
tion with (i) torpedo power plants and (ii) the general 
problem of power-plant design. 

OEMsr-1289 

Massachusetts Institute of Technology 
Cambridge, Mass. 

Conduct studies and experimental investigations in connec- 
tion with new and improved fuels for torpedoes, including 
survey of power supplies for jet-propelled missiles. 

OEMsr-1342 

Newark College of Engineering 

Newark, New Jersey 

Conduct studies and experimental investigations in connec- 
tion with a development and test program for Navy 
Project NO-176. 




155 


SERVICE PROJECT NUMBERS 


The projects listed below were transmitted to the Executive Secretary, 
XDRC, from the War or Navy Department through either the War 
Department Liaison Officer for NDRC or the Office of Research and 
Inventions (formerly the Coordinator of Research and Development), 
Navy Department. 


Service Project umber Subject 


NO-176 

NO-236 

N-121 


Torpedoes for high-speed aircraft 
Investigation of torpedo fuels 
Torpedo survey 


INDEX 


The subject indexes of all STR volumes are combined in a master index printed in a separate volume. 

For access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. 


Aerodynamic constants of torpedoes 
see ^lark 13 torpedo, aerodynamic 
constants 

Ailerons, gyro-controlled (roll stabi- 
lizer), 37 
Air trajectory 

see Aircraft torpedoes, theory of flight 
Aircraft torpedoes, 10-15 
see also Mark 13 torpedo 
air travel, 10-1 1 * 
breakable wood tail, 10 
control of underwater run, 13-15 
requirements, 4, 6 
speed and range, 4 
steering accuracy, 4 
water entry, 1 1-13 

Aircraft torpedoes, theory of flight, 
21-49 

comparison with flight in vacuum, 
34 

damping coefficient, 10, 42-44 
drag coefficient, 44 
effect of drag ring, 34, 45 
effect of stabilizers, 24 
equations of motion, 21-25 
horizontal distance and velocity, 21, 
27 

lift coefficient, 24 

moment coefficient, 19-20, 22, 42-45 
pitch oscillations, 24-25, 30-32, 38-40 
probable calculation errors, 38 
release conditions, 32-36 
roll, 37, 40 

torpedo center of gravity, 26-27 
trajectory angle, 28-30 
vertical fall and velocity, 21, 26, 27, 
41-42 

wind, 46-49 

yaw oscillations, 25, 36-37 
American Can Company, 1 38 
Angular velocity in air, 32 
Angular velocity in water, 60-68, 76- 
77 

entry velocity, 11 
formulas, 76 
nose shape, 60-65 
torpedo .stability, 118-1 19 
whip of torpedoes, 54, 61-67, 76 
yaw, 76 

Automobile torpedo 
definition, 3 

predetermined course and depth, 7-8 
speed, 3 

Axial velocity, 59-60 


Blackwell, diameter of water cavity, 73 
British research 

M.A.T. IV torpedo stabilizer, 10, 45 
rebounding effect of torpedo, 99 
torpedo stabilizer, 10, 45 
Broaching of torpedoes, 1 12-113 

California Institute of Technology Tor- 
pedo Launching Range, 52 
Cavitation, 68-106 
cavity contraction, 97 
cavit}^ diameter, 73 
deep closure, 84-86 
definition, 100 
drag coefficient, 75, 101 
formula for cavity shape, 84 
initial phase, 11-12 
kinematic theory of cavity shape, 
73-74 

nose contact with water, 68-84 
position in cavity, 89 
radius of cavity after impact, 74 
recommendations for future research, 
84, 103 

shroud ring, 112 

studies using scaled models, 103-104 
surface closure, 86-87 
tail forces, 87, 99, 105-106 
time of clo.sure, 85 
trajectory angle at cavity, 97 
transition region, 99-103 
water entry of torpedo, 84-106 
whip-producing stage, 104-105 
Columbia University, pneumatic con- 
trol, 138 

Control mechanisms, 7-8, 13-15, 123- 
148 

dependence on torpedo hydrodynam- 
ics, 7 

depth control, 4-5, 14-15, 127-128, 
143-148 

effect on trajectory, 110-111, 126-127 
pneumatic control, 104, 138-139 
proportional control, 123-125, 129- 
134, 138-139, 146 
purpose, 123 
requirements, 7-8 
roll control, 106-107, 127-128 
steering control, 4-5, 14, 127-128, 
138-139 

two-position control, 125-127, 135- 
137 

types, 123 

Cross force, 20, 98-99 


Damping coefficient in air 
drag ring, 43-44 
Mark 13 torpedo, 43-44 
pitching motion, 34 
propeller, 42 
stabilizers, 10, 43-44 
Depth control, 14-15, 143-148 
depth engine, 146-147 
effect of roll, 127-128 
effect of shroud ring, 112 
effect on steering, 127-128 
effect on trajectory, 12 
gyroscope control, 15, 146 
influence exploders, 4 
limitations of hydrostatic bellows, 
143 

pendulum, 15, 143-148 
principles and methods, 4-5, 143-147 
proportional control, 104, 123-125, 
129-134, 138-139 
reduction of phase lags, 147 
requirements, 4-5 

transmission of signal to elevators, 
146 

Destroyer-launched torpedoes, 4 
Drag coefficient 

as function of cavitation parameter, 
75, 101 

as function of pressure distribution, 
75-76 

at water entry, 12 
effect of head shape, 95-96 
effect of nose shape, 74-75 
for Mark 13 torpedo, 44 
for rockets, 96 
formulas, 20, 74 
in air, 25-26, 44 

measured by potential flow method, 
74-75 

of propellers, 95 
of shroud ring, 95 
of torpedo nose, 11, 74-75, 84, 96 
of torpedo tail, 85, 94-96 
Drag ring, 43-45, 52-58 

damping moment coefficient, 43-44 

effect on air trajectory, 34 

Markl; 10, 43-45 

peak deceleration, 58 

reduction of elastic wave pressure, 55 

water impact, 45, 52-53 

Equations of motion in air, 21-24 
moment force, 22 
total velocity, 23 


157 


158 


INDEX 


trajectory affected by weight and 
drag, 25 

vertical and horizontal velocity, 21 
yawing motion, 25 
Equations of motion in water 
see Motion equations in water 
Explosive charge, torpedoes, 3, 6 

Fins, effect on torpedo roll, 91-92, 94 
Flow-forming stage, 53-68 
angular velocity, 60-68 
damage to torpedo, 58 
duration, 54-55 
flow separation, 68, 73 
impulsive axial velocity change, 59- 
60 

nose dimensions, 54, 59 
orientation of torpedo axis, 63-65 
pitch angle, 64 

pressures acting on torpedo nose, 
55-58 
ricochet, 67 
yaw angle, 64 
Formulas 

air velocity, 23 
angular velocity in water, 76 
area of pressure at point of water im- 
pact, 51-52 
cavity shape, 84 
cross force, 20 
drag coefficient, 20, 74 
equations of motion, air flight, 21-24 
equations of motion, underwater run, 
116-119 

lift coefficient, 98 

longitudinal force due to pressure, 
52 

magnitude of torpedo heel, 119 
moment coefficient, 19, 22 
motion along trajectory, 87 
motion in vertical plane, 87 
pitch angle, 25, 31, 64, 88 
pitching moment of torpedo, 25 
pressure on torpedo nose, 55, 68-70 
radius of cavity, 74 
roll angle, 37 
speed and range, 8 

time duration of flow-forming state, 
54 

time of cavity closure, 85 
trajectory angle at cavity, 97 
water entry velocity, 51 
whip after how-forming stage, 76 
yaw angle, 25, 64 
Foxboro Corporation, 146 

German tor[)edo (Kurt), 67 
Glide bombing, 34 

GOR (gyroscopic orientation n'corder), 
83 

Gyro-controlled ailerons (stabilizer), 37 


Gyroscope, use in torpedoes 
depth control, 15, 146 
steering control, 138 

Heel of torpedo, 119-120 
Hurwitz criterion, stability of con- 
trolled torpedoes, 129-130 
HVAR rocket, drag coefficient, 96 
Hydrodynamic forces 
see also Drag coefhcient 
cross force, 20, 98-99 
damping moment and force, 20 
effect of propellers, 98-99, 115 
function of attack angle, 114 
lift coefficient, 96-98 
moment, 19-20 

recommendations for future research, 
115 

tail forces, 94-99 
torpedo stability, 13-14 
underwater run of torpedo, 114-116 
Hydropressure plugs for pressure mea- 
surements, 55 

Hydrostatic bellows for torpedo depth 
control, 143 

Inhuence exploders for torpedo depth 
control, 4 

JMA 2 rockets, drag coefficient, 96 

Kurt (German torpedo), 67 

Langley Field, torpedo stabilizer tests, 
42 

Lift coefficient 

effect on torpedo trajectory, 24 
formula, 98 

Mark 13 torpedo, 44, 98 
propellers, 99 
recovery stage. 111 
torpedo tail, 96-98 
Limited control of torpedoes 

see Two-position control of torpedoes 
Idnear control of torpedoes 

see Proportional control of torpedoes 

Mark 1 drag ring, 10 

aerodynamic constants, 43-45 
damping force coefficient, 43-44 
effect on air trajectory, 45 
improvement of water entry, 45 
Mark 2 torpedo stabilizer 

aerodynamic constants, 43-45 
damping coefficient, 10, 43-44 
moment coefficient, 43 
pitching motion, 40 
Mark 7 rocket, drag coefficient, 96 
Mark 13 torpedo 

see also Aircraft torpedoes, theory of 
high! 


control mechanisms, 125-137 
duration and area of pressure at point 
of impact, 51-52 
head shape, 50 

hydrodynamic constants, 98, 115 
inertia parameters, 129-130 
shroud ring, 108 
stabilizer, 10, 40, 43-45 
terminal velocity, 27 
yawing motion, 36-37 
Mark 13 torpedo, aerodynamic con- 
stants, 42-45 

damping coefficient, 43-44 
drag coefficient, 44 
effect of drag ring, 43 
effect of propeller, 42 
effect of shroud ring, 44-45 
lift coefficient, 44 

method of obtaining aerodynamic 
constants, 42-43 
moment coefficient, 42-44 
use of strip camera photographs, 42 
Mark 25 torpedo, pneumatic steering 
control, 138-139 

M.A.T. IV torpedo stabilizer, 10, 45 
MBB torpedo dummy, roll tests, 92-94 
Models, scaled, cavitation studies, 103- 
104 

Moment coefficient in air, 19-20, 42-43 
damping moment, 43-44 
effect of stabilizer, 43, 45 
formula, 19, 22 
horizontal plane, 44 
Mark 13 torpedo, 42-44 
Motion equations in air, 21-24 
moment of force, 22 
total velocity, 23 

trajectory affected by weight and 
drag, 25 

vertical and horizontal velocity, 21 
yawing motion, 25 
Motion equations in water, 106-111 
assumptions, 106-107 
criterion of dynamic stability, 118- 
119 

horizontal plane, 107-108 
nose forces, 77-79 
recovery stage, 106-111 
roll, 106-109 

steering equations, 139-142 
underwater run, 1 16-119 
vertical i)lane, 108-110 
with controls, 1 10-1 1 1 

Newport Torpedo Station, 42 
Nose forces under water, 68-84 
damage to torpedo, 83-84 
drag coefficient, 74-75 
elastic pressure wave, 52, 55 
equal ions of motion, 77-79 
how separation, 68, 73 




INDEX 


159 


kinematic theory of cavity shape, 
73-74 

nose cap, 52-53, 55 
nose-down lift, 104 
pressure distribution, 55-58, 68-72 
recommendations for future research, 
56, 67-68, 84 
venting, 104-105 
water impact, 52-53, 83-84 
Nose shape 

effect on angular velocity, 60-67 
effect on drag coefficient, 11, 74-75, 
84, 96 

effect on flow separation, 54, 59 
effect on water entry, 12 
hemispherical nose, 61-64, 74-75 
Xyquist criterion,' torpedo stability, 
130-134 

advantages, 133-134 
pendulum-hydrostat proportional 
system, 147-148 

Optical whip recorder, 62 

Pallet mechanism for steering torpe- 
does, 138 

Pendulum for torpedo depth control, 
15, 143-146 

antiresonance frequency, 145 
captured pendulum, 146 
disadvantages, 146 
equation of motion, 144-145 
limitations, 15 

proportional control, 147-148 
Photography of torpedo aerodynamics, 
42 

Pickel barrel 
see Drag ring 

Pitching motion in air, 24-25, 30-36, 
38-40 

angular velocity, 32 
damping effect, 34 
definition, 25 
effect of roll, 40 

effect of torpedo stabilizer, 24, 40 
effect of wind, 47-48 
formula, 25, 31, 64 
frequency, 30 

function of altitude and time, 38 
positive and negative pitch angles, 30 
probable calculation error, 39-40 
release conditions, 32 
zero pitch angle, 24 
Pitching motion in water, 78-79 
critical angle, 12, 78 
effect of torpedo design, 78-79 
effect of whip at entry, 78 
flow-forming stage, 64 
formula, 88 
shallow water, 78 
steady state angle, 119 


Pneumatic steering control, 104, 138-139 
Pressure acting on torpedo, 50-52 
area at point of water impact, 51-52 
duration at point of water impact, 
51-52 

elastic pressure wave, 52, 55 
transverse and longitudinal force, 52 
Pressure measurements on torpedo 
nose, 68-72 

approximate method for a sphere, 72 
flow-forming stage, 55-58 
formula, 68-70 
h 3 ^dropressure plugs, 55 
noses with discontinuities, 72 
point of flow separation, 73 
potential flow method, 68-72 
semiempirical method, 72 
sphere, 68-71 
variational method, 72 
Propellers 

aerodynamic constants, 42 
damping coefficient, 42, 95 
hydrodynamic constants, 98-99, 115 
lift coefficient, 99 
roll velocity, 91, 93 

Proportional control of torpedoes, 14, 
123-125, 129-134, 138-139 
analogy with linear feed-back ampli- 
fier, 124 

criterion for stability, 118, 123-125, 
129-134, 147-148 
definition, 123 

force-proportional control, 139, 146 
pendulum-hydrostat control, 147-148 
pneumatic control, 104, 138-139 
Propulsion mechanism, energy require- 
ments, 6-7 

Range of torpedoes, 4, 8 
Recommendations for future research 
cavity shape formed by torpedo, 84, 
103 

force coefficients during open cavity 
and transition stage, 103 
hydrodynamic constants of torpe- 
does, 115 

torpedo nose forces, 56, 67-68, 84 
Recovery stage, 106-1 13 
broaching, 112-113 
effect of shroud ring, 112 
equations of motion, 106-111 
lift force. 111 

motion in recovery stage, 1 1 1-113 
Ricochet of torpedoes under water, 67 
Roll in air, 37, 40 
Roll in water, 91-94 

controls, 106-107, 127-128 
distortion of trajectory, 94 
effect of fins, 91-92 
effect of metacentric height, 91 
effect of propellers, 91 


effect on steering control, 127-128 
equations of motion, 106-109 
tests with torpedo dummy, 92-94 
yawing motion, 91, 92 

Scaled models, cavitation studies, 103- 
104 

Shroud ring 

aerodynamic constants, 44-45 
broaching, 112-113 
depth of dive, 112 
drag coefficient, 95 

dynamic stability of torpedo, 141, 142 
effect after cavity collapse, 112 
inertia, 129-130 
phase lag, 137 

Specifications for torpedoes, 3-9 
aircraft torpedoes, 4, 6 
control mechanisms, 4-5, 7-8 
explosive charge, 3, 6 
external shape, 6 
length of underwater run, 3 
propulsion mechanism, 6-7 
submarine- and destroyer-launched 
torpedoes, 4 
weight and size, 8-9 
Speed of torpedoes 

aircraft, 4, 23, 25-27, 41-42 
angular velocity, 60-68, 76-77 
axial velocity, 59-60 
submarine and destroyer launched, 4 
terminal velocity, 27 
water entry velocity, 51, 83-84 
Spoiler rings for torpedo nose, 6 
Stability of torpedoes 

criterion of dynamic stability, 118, 
123-125 

effect of shroud ring, 141-142 
effect on angular velocity, 118-119 
yawing motion, 1 18-1 19 
Stabilizers for torpedoes, 10-11 
British M.A.T. IV; 10, 45 
damping effect, 10, 43-44 
drag ring, 10, 34, 43-44, 52-58 
effect on moment, 43, 45 
effect on pitching motion, 24, 40 
effect on trajectory, 24 
gyro-controlled ailerons, 37 
Hurwitz criterion of stability, 129-130 
Mark 1 drag ring, 10 
Mark 2; 10, 40, 43-45 
Nyquist criterion of stability, 130- 
134, 147-148 

Steering control of torpedoes, 138-139 
accuracy, 4 

approach to turn, equations of mo- 
tion, 139-141 
effect of roll, 128 
effect on depth-keeping, 127 
methods, 138-139 
}mllet mechanism, 138 


‘TWTPIOiMlillMnU. 


160 


INDEX 


pneumatic control, 104, 138-139 
proportional mechanism, 14, 123- 
125, 129-134, 138-139 
pull-out from turn, equations of mo- 
tion, 141-142 
requirements, 4-5 
roller attached to gyro gimbal, 138 
two-position mechanism, 14, 123, 
125-127, 135-137 

Strip photography, use in obtaining 
aerodynamic constants, 42 
Submarine-launched torpedoes 
shape requirements, 6 
speed and range, 4 
steering accuracy, 4 

Tail forces under water, 87 

behavior and trajectory after tail 
slap, 105-106 
cavity contraction, 97 
drag coefficient, 85, 94-96 
lift coefficient, 96-98 
motion along trajectory, 87 
motion in horizontal plane, 88 
motion in vertical plane, 87-88 
oscillatory trajectory, 90 
position of torpedo at tail slap, 79-83 
roll in cavity, 91-94 
trajectory angle at cavity closure, 97 
Terminal velocity, definition, 27 
Torpedo power plant, 149 
Torpedoes 

see Aircraft torpedoes, theory of flight; 
Cavitation; Flow-forming stage; 
Recovery stage; Trajectory un- 
der water; Water impact stage 
Toss bombing, 34 

Townend extensions for torpedo nose, 6 
Trajectory in air 

see Aircraft torpedoes, theory of flight 
Trajectory under water, 114-120 

effect of control mechanism, 12, 126- 
127 


effect of heel, 1 19-120 
equations of motion and stability, 
116-119 

estimates of hydrodynamic coeffi- 
cients, 114-116 
initial phases, 50 
motion along trajectory, 87 
nose contact with water, 76-84 
ricochet, 67 

rolling motion, 91-94, 106-109 
steady circling, 119-120 
steady running, 119 
Transition region, 99-103 

cavitation parameter, 101-102 
drag coefficient, 101 
extent of transition region, 99-102 
noncavitating motion, 102 
Two-position control of torpedoes, 125- 
127, 135-137 

analogy to low-pags amplifier, 125 

definition, 123 

phase lag, 125, 127, 135, 147 

response of system, 125-126 

stability, 123, 125 

time lag, 14 

Underwater trajectory 

see Trajectory under water 
United Shoe r^Iachinery Corporation, 
146 

University of Michigan, 42 

Venting of torpedoes, 104-105 

Water entry, 11-13, 50-84 
angular velocity, 1 1 
cavity formation, 11-12 
critical pitch angle, 12 
drag force, 12 
effect of drag ring, 45 
effect of nose shape, 12 
effect of wind, 12-13 
entry conditions, 50 


flow-forming stage, 53-68 
nose contact with water, 68-84 
phases of initial underwater trajec- 
tory, 50 

Water impact stage, 50-53 

effect of drag ring and nose cap, 45, 
52-53 

pressure, 50-52 

Westinghouse Electric and Manufac- 
turing Company 
captured pendulum, 146 
force-proportional electric depth con- 
trol, 146 

steering control for torpedoes, 138 
two-position electric torpedo control, 
147 

Whip of torpedoes 
definition, 53 

effect of nose shape and size, 66- 
67 

flow-forming stage, 61-64, 76, 104- 
105 

forces producing the whip, 65-66 
optical whip recorder, 62 
Wind, effect on torpedo trajectory, 
46-49 

distribution of wind velocity, 39 
pitch angle, 47-48 
trajectory angle, 47 
water entry, 12-13 
yaw angle, 48-49 
Wright Field, 42 

Yawing motion 
definition, 25 

dynamic stability of torpedo, 118- 
119 

effect of wind, 48-49 
effect on angular velocity, 76 
effect on roll velocity, 91, 92 
equations of motion, 25 
flow-forming stage, 64 
Mark 13 torpedo, 36-37 


















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